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PSLE

PSLE Foundation Mathematics

<p>A particle $P$ moves in a straight line so that&nbsp;$t$ seconds after leaving a fixed point $O$, its velocity $v$ ms${}^{-1}$ is given by $v = (2t - 3)^2 - 9$.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Sketch the $v - t$ graph of the particle $P$ for&nbsp;$0 \le t \le 5$.</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Hence or otherwise,</div><ol class="i-list"><li data-list-item-id="item-b-i" data-list-item-path="item-b/item-b-i"><div class="list-item-paragraph">find the range of values of $t$ for which the acceleration of $P$ is less than $4$ m/$\mathrm{s}^2$.</div></li><li data-list-item-id="item-b-ii" data-list-item-path="item-b/item-b-ii"><div class="list-item-paragraph">find the distance travelled by $P$ in the first $5$ seconds.</div></li></ol></li></ol>

A particle $P$ moves in a straight line so that $t$ seconds after leaving a fixed point $O$, its velocity $v$ ms${}^{-1}$ is given by $v = (2t - 3)^2 - 9$.

Sketch the $v - t$ graph of the particle $P$ for $0 \le t \le 5$.

Interpreting and Sketching Motion Graphs

Differentiating Displacement and Velocity

Distance Travelled and Average Speed

Motion Conditions, Rest and Direction

Kinematics

<p>Mr. Tan drove from home to Changi Airport to pick up his friend. On his way to the airport, he stopped by at a petrol station to refuel his car. After arriving at the airport, Mr. Tan waited for his friend. Then they went to a restaurant to have dinner. The distance-time graph below shows Mr. Tan’s entire journey till he reached the restaurant. You may assume all locations lie on a straight line.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767606405465-1vcek9hh1or.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2MDY0MDU0NjUtMXZjZWs5aGgxb3IucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjc4OTI2MywiZXhwIjoxNzkyNzg5NTYzfQ.RarxkCoQ460VvOYAI2ezlamsLOXFSwQI8Aui1Xun07Q" style="max-width: min(280px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the distance from the restaurant to Mr. Tan’s home.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find, in minutes, the time Mr Tan spent</span></div><ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">refuelling his car,</span></div></li><li><div class="list-item-content"><span class="list-item-text">waiting for his friend at the airport.</span></div></li></ol></li><li><div class="list-item-content"><span class="list-item-text">Find the gradient of each of the following line segments, and state clearly what each gradient represents.</span></div><ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">$OA$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">$AB$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">$BC$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">$DE$.</span></div></li></ol></li></ol>

Mr. Tan drove from home to Changi Airport to pick up his friend. On his way to the airport, he stopped by at a petrol station to refuel his car. After arriving at the airport, Mr. Tan waited for his friend. Then they went to a restaurant to have dinner. The distance-time graph below shows Mr. Tan’s entire journey till he reached the restaurant. You may assume all locations lie on a straight line.

Find the distance from the restaurant to Mr. Tan’s home.

Find the gradient of each of the following line segments, and state clearly what each gradient represents.

 Linear Graphs in Real Life Problems

Functions and Graphs

<p>The diagram shows the speed-time graph of two cars $A$ and $B$ during a period of $20$ seconds.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767347348016-pvrx55winz.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjczNDczNDgwMTYtcHZyeDU1d2luei5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyNzg5MjYzLCJleHAiOjE3OTI3ODk1NjN9.G_C8q1yCrax0P2dBT3WFN1R9_30YFjbzxO7S6U7upzk" style="max-width: min(300px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>Calculate</p>
<ol class="abc-list"><li><div class="list-item-paragraph">the acceleration of car $A$ during the first $6$ seconds of its motion,</div></li><li><div class="list-item-paragraph">the average speed of car $A$ during the $20$ seconds of its motion,</div></li><li><div class="list-item-paragraph">after how many seconds will car $A$ travel at the same speed as car $B$.</div></li></ol>

The diagram shows the speed-time graph of two cars $A$ and $B$ during a period of $20$ seconds.

Kinematics Analysis: Interpreting Speed-Time and Acceleration-Time Graphs

<ol class="abc-list"><li><div class="list-item-paragraph">A car accelerates uniformly from rest until it reaches a speed of $30$ m/s in $20$ s.</div><div class="list-item-paragraph">Calculate the acceleration of the car.</div></li><li><div class="list-item-paragraph">The car continues to accelerate but at a slower rate for a further $30$ s until it reaches a speed of $40$ m/s. </div><div class="list-item-paragraph">Then the car begins to slow down with uniform retardation of $2.5$ m/s$^{2}$.</div><ol class="i-list"><li><div class="list-item-paragraph">Draw the speed-time graph for the first $50$ seconds of the journey.</div><div class="content-block list-item-image" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767346033745-ocf5jbrasc8.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjczNDYwMzM3NDUtb2NmNWpicmFzYzgucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjc4OTI2MywiZXhwIjoxNzkyNzg5NTYzfQ.N6YETNsy0ljBkOCOIbqkjZ1VxKQWtxo3nDMEfoa9v7w" alt="Sub-item image" class="aligncenter custom-width" style="max-width: min(250px, 100%); height: auto;" /></div></li><li><div class="list-item-paragraph">Calculate the distance it travels during the period from $t=20$ to $t=50$.</div></li><li><div class="list-item-paragraph">Calculate how long it would take for the car to come to rest.</div></li><li><div class="list-item-paragraph">On the diagram in the answer space below, sketch the acceleration-time graph of the car for the entire journey.</div><div class="list-item-paragraph">Indicate clearly the values on the vertical and horizontal axes.</div></li></ol><div class="list-item-after-content" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767346111987-01np3rk7a4h1.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjczNDYxMTE5ODctMDFucDNyazdhNGgxLnBuZyIsInNjb3BlIjoiZG93bmxvYWQiLCJpYXQiOjE3OTI3ODkyNjMsImV4cCI6MTc5Mjc4OTU2M30.GCkZm2sCRdD0084byyQttIVJQiyvT43M6mNwgIHmUTc" alt="After sub-items image" class="aligncenter custom-width" style="max-width: min(350px, 100%); height: auto;" /></div></li></ol>

Kinematics Analysis: Motion of a Car via Speed-Time and Acceleration-Time Graphs

<p>The diagram shows the speed-time graph of two objects $A$ and $B$ for the first $10$ seconds of their motion.  </p>
<p>Object $A$ travels at a constant speed of $12$ m/s throughout the $10$ seconds. </p>
<p>Object $B$ starts from rest, attains a speed of $18$ m/s after $4$ seconds and then travels at a constant speed.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767345013007-oymhia6d5xn.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjczNDUwMTMwMDctb3ltaGlhNmQ1eG4ucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjc4OTI2MywiZXhwIjoxNzkyNzg5NTYzfQ.VN5kMKkFnQAm9LmyGKLE90r7iVT_XfP362bwXTbCelk" style="max-width: min(350px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-paragraph">Calculate</div><ol class="i-list"><li><div class="list-item-paragraph">the distance traveled by object $B$ during the first $4$ seconds of its motion.</div></li><li><div class="list-item-paragraph">the average speed of object $B$ for the first $10$ seconds of its motion.</div></li><li><div class="list-item-paragraph">the time taken for object $B$ to travel at the same speed as object $A$.</div></li><li><div class="list-item-paragraph">the time taken for both to travel the same distance.</div></li></ol></li><li><div class="list-item-paragraph">After $10$ seconds, both objects slow down at the same rate until they come to rest.</div><div class="list-item-paragraph">Given that object $A$ comes to rest after a further $6$ seconds, calculate the</div><ol class="i-list"><li><div class="list-item-paragraph">deceleration of object $A$.</div></li><li><div class="list-item-paragraph">total time taken for object $B$ to travel the whole journey.</div></li></ol></li></ol>

The diagram shows the speed-time graph of two objects $A$ and $B$ for the first $10$ seconds of their motion.  

Object $A$ travels at a constant speed of $12$ m/s throughout the $10$ seconds. 

Object $B$ starts from rest, attains a speed of $18$ m/s after $4$ seconds and then travels at a constant speed.

Analysis of Speed-Time Graphs: Constant and Uniformly Accelerated Motion

<p>The diagram below shows the speed-time graph of a car which starts from rest.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767344455189-3zp41azjff1.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjczNDQ0NTUxODktM3pwNDFhempmZjEucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjc4OTI2MywiZXhwIjoxNzkyNzg5NTYzfQ.TLxcfzMq-iVNPFgZjxOIdoUtGYo2R2C75bZ962S87Co" style="max-width: min(300px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-paragraph">Calculate the speed of the car at $10$ seconds.</div></li><li><div class="list-item-paragraph">Find the value of $u$, if the total distance covered in the first $40$ seconds is $1000$ m.</div></li><li><div class="list-item-paragraph">Draw the distance-time graph of the car in the axes below.</div></li></ol>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767344507217-5ju360ci69a.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjczNDQ1MDcyMTctNWp1MzYwY2k2OWEucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjc4OTI2MywiZXhwIjoxNzkyNzg5NTYzfQ.owQ_5dpy76PYkxQO78qEu97-kDumUXBHkxMrVxFNG7g" style="max-width: min(300px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>

The diagram below shows the speed-time graph of a car which starts from rest.

Analyzing the Speed-Time Graph of a Car

<p>Alice participated in a $30$ km run last month.  </p>
<p>The distance-time graph below represents her run from the start to the end.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767339093297-dh23shr22ns.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjczMzkwOTMyOTctZGgyM3NocjIybnMucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjc4OTI2MywiZXhwIjoxNzkyNzg5NTYzfQ.Yi1o9q7CvdYhpjPFXDedMpQsAiXBw1sT6mkfUpkvhAE" style="max-width: min(500px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-paragraph">Using Alice’s speed, describe her run as shown in the graph above.</div></li><li><div class="list-item-paragraph">What is Alice’s speed at $5$ a.m.?</div></li><li><div class="list-item-paragraph">Find the average speed of Alice’s run.</div></li><li><div class="list-item-paragraph">Bella also took part in the same run as Alice.  She started at the same time, ran at a constant speed and completed her run at $0736$ h.</div><ol class="i-list"><li><div class="list-item-paragraph">Draw and label Bella’s graph on the given distance-time graph for Alice below.</div></li><li><div class="list-item-paragraph">At what time did Alice overtake Bella?</div></li></ol></li></ol>

Alice participated in a $30$ km run last month.  

The distance-time graph below represents her run from the start to the end.

Estimating Gradients from Tangents

Alice's 30 km Run: Analyzing Distance and Speed

<p>The diagram below shows the distance-time graph for the first $20$ seconds of a journey.</p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--1-2" data-layout="1-2"><div class="question-column" style="flex: 1; align-self: flex-start;"><p>Find</p><ol class="abc-list"><li><div class="list-item-paragraph">the speed at $t=2$.</div></li><li><div class="list-item-paragraph">the average speed from $t=12$ to $t=20$.</div></li><li><div class="list-item-paragraph">the acceleration when&nbsp;$t=4$.</div></li></ol></div><div class="question-column" style="flex: 2; align-self: flex-start;"><div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1766400571568-gzm3gclk4ii.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjY0MDA1NzE1NjgtZ3ptM2djbGs0aWkucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjc4OTI2MywiZXhwIjoxNzkyNzg5NTYzfQ.hc2oiw0wfZslZqjTt13UJ0_4MxCFvAVAr0U262JEl_8" style="max-width: min(500px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div></div></div></div>

The diagram below shows the distance-time graph for the first $20$ seconds of a journey.

Distance-Time Graph Analysis for a Journey

<p>The graph shows the journeys of Andy and Gary between Towns $P$ and $Q$ which are $25$ km apart.	</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1766398669290-v3f972fufzn.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjYzOTg2NjkyOTAtdjNmOTcyZnVmem4ucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjc4OTI2MywiZXhwIjoxNzkyNzg5NTYzfQ.TUm5SIlo_n6kYpbXxJOxNxIVnFvTqvs5WdlxuAWaHsc" style="max-width: min(500px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>Using the graph,</p>
<ol class="i-list"><li><div class="list-item-paragraph">find the distance Gary was from Town $Q$ when he stopped for a rest,</div></li><li><div class="list-item-paragraph">find the amount of time Gary rested,</div></li><li><div class="list-item-paragraph">state the time Andy overtook Gary,</div></li><li><div class="list-item-paragraph">find the distance between Gary and Andy at $1050$,</div></li><li><div class="list-item-paragraph">calculate the average speed for Andy’s whole journey, correct to $3$ significant figures.</div></li></ol>

The graph shows the journeys of Andy and Gary between Towns $P$ and $Q$ which are $25$ km apart.	

Interpreting Distance–Time Graphs of Two Travellers

<p>Match the distance-time graph that best represents the situations described below</p>
<ol class="abc-list"><li data-list-item-id="item-1766392975353-83dbad16q" data-list-item-path="item-1766392975353-83dbad16q"><div class="list-item-paragraph">A cyclist starts from rest and accelerates uniformly.  </div><div class="list-item-paragraph">He then maintains a constant speed.</div></li><li data-list-item-id="item-1766392977737-f99v47w5q" data-list-item-path="item-1766392977737-f99v47w5q"><div class="list-item-paragraph">A bean bag is thrown upward into the air.</div></li><li data-list-item-id="item-1766392978533-x2ib5djit" data-list-item-path="item-1766392978533-x2ib5djit"><div class="list-item-paragraph">A ball is released at the top of a ramp and allowed to roll until it stops.</div></li></ol>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1766393035706-hiaub0ckefh.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjYzOTMwMzU3MDYtaGlhdWIwY2tlZmgucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjc4OTI2MywiZXhwIjoxNzkyNzg5NTYzfQ.1ugrUJLGOviopX5Xo2P0ifQAndZmPQzeaygXM5UffRo" style="max-width: min(600px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>In these sketches, “Distance” means position or height relative to a reference point where it decreases; it cannot mean cumulative distance travelled. The shapes are qualitative sketches, not exact physical models.</p>

Match the distance-time graph that best represents the situations described below

In these sketches, “Distance” means position or height relative to a reference point where it decreases; it cannot mean cumulative distance travelled. The shapes are qualitative sketches, not exact physical models.

Matching Motion Descriptions to Distance–Time Graphs

<p>The diagram below shows the distance-time graph for the first $18$ seconds of a journey.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1765981769356-24ffx73okv9.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjU5ODE3NjkzNTYtMjRmZng3M29rdjkucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjc4OTI2MywiZXhwIjoxNzkyNzg5NTYzfQ.mkKZUj69pyCgOMnjyZ_HY_34Nv-6d8B-ZJRothliHBc" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the average speed during the first $8$ seconds.</div></li><li><div class="list-item-paragraph">Find the speed in the last $4$ seconds.</div></li><li><div class="list-item-paragraph">Find the acceleration when&nbsp;$t=16$ seconds.</div></li><li><div class="list-item-paragraph">The speed increases uniformly from $2\mathrm{ m }{{\mathrm{s}}^{-1}}$ when&nbsp;$t=0$s, to $t=8$s.  </div><div class="list-item-paragraph">Sketch the speed time graph for the journey from $t=0$s to $t=18$s.</div></li></ol>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1765981959161-xbsnngwiaw.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjU5ODE5NTkxNjEteGJzbm5nd2lhdy5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyNzg5MjYzLCJleHAiOjE3OTI3ODk1NjN9.eMnaU26_f2YtQC1nGNyXHYafgC-jD0uUpm6a575-8oE" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>

The diagram below shows the distance-time graph for the first $18$ seconds of a journey.

Conversion of Distance-Time Graph to Speed-Time Graph

<p>A truck travels at a constant speed of $20$ m/s for&nbsp;$10$ seconds.  </p>
<p>It then decelerates at a constant rate to rest in ten seconds.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find</div><ol class="i-list"><li><div class="list-item-paragraph">the acceleration of the truck during the last $10$ seconds,</div></li><li><div class="list-item-paragraph">the total distance travelled by the truck in the $20$ seconds.</div></li></ol></li><li><div class="list-item-paragraph">On the axes below, draw</div><ol class="i-list"><li><div class="list-item-paragraph">the speed-time graph for the whole journey of the truck,</div><div class="content-block list-item-image" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1765980945463-gd0qq5gomv6.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjU5ODA5NDU0NjMtZ2QwcXE1Z29tdjYucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjc4OTI2MywiZXhwIjoxNzkyNzg5NTYzfQ.Wt8spOsPDAuqyV083HA8DDYoRsO_vooIGR8BjfLKraQ" alt="Sub-item image" class="aligncenter custom-width" style="max-width: min(300px, 100%); height: auto;" /></div></li><li><div class="list-item-paragraph">the distance-time graph for the whole journey of the truck.</div><div class="content-block list-item-image" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1765980984678-r66xeb25c69.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjU5ODA5ODQ2NzgtcjY2eGViMjVjNjkucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjc4OTI2MywiZXhwIjoxNzkyNzg5NTYzfQ.yCpjaHO7esjHRh5qVZOu6cHx4gY1PrT2atSGpSK5D8c" alt="Sub-item image" class="aligncenter custom-width" style="max-width: min(300px, 100%); height: auto;" /></div></li></ol></li></ol>

A truck travels at a constant speed of $20$ m/s for $10$ seconds.  

It then decelerates at a constant rate to rest in ten seconds.

Conversion of Speed-Time Graph to Distance-Time Graph

<p>The diagram shows the distance-time graph of an object as it moves.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1765980456237-nqp78uxp7gl.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjU5ODA0NTYyMzctbnFwNzh1eHA3Z2wucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjc4OTI2MywiZXhwIjoxNzkyNzg5NTYzfQ.T6tdH9wXbO6xoRn7fNdzuoOgEAbCjg0OXWpR6uv_WsA" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li data-list-item-id="item-1765980467336-4l1j6ndbd" data-list-item-path="item-1765980467336-4l1j6ndbd"><div class="list-item-paragraph">Write down the speed of the object at $7$ seconds.</div></li><li data-list-item-id="item-1765980467908-b073d7puh" data-list-item-path="item-1765980467908-b073d7puh"><div class="list-item-paragraph">After $10$ seconds, the speed of the object is $0.75$ m/s.  Calculate the value of $k$.</div></li><li data-list-item-id="item-1765980469470-jay17a88k" data-list-item-path="item-1765980469470-jay17a88k"><div class="list-item-paragraph">Given that the object accelerates at a constant rate, calculate the speed of the object at $5$ seconds.</div></li><li data-list-item-id="item-1765980470192-9fmy5sjiy" data-list-item-path="item-1765980470192-9fmy5sjiy"><div class="list-item-paragraph">Complete the speed-time graph of the object in the diagram below.</div></li></ol>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1765980525606-afjxp64w5yi.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjU5ODA1MjU2MDYtYWZqeHA2NHc1eWkucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjc4OTI2MywiZXhwIjoxNzkyNzg5NTYzfQ.NagERRezxJH6YfGG9ATMo5HDE6zj6RivK7nv-qkWEZ0" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>For (c) and (d), the shown curve is interpreted as having a horizontal tangent at the start, so the object starts from rest. Without that interpretation, uniform acceleration and a distance of $4$ m in $5$ s give only $u+v=1.6$, not a unique final speed.</p>

The diagram shows the distance-time graph of an object as it moves.

For (c) and (d), the shown curve is interpreted as having a horizontal tangent at the start, so the object starts from rest. Without that interpretation, uniform acceleration and a distance of $4$ m in $5$ s give only $u+v=1.6$, not a unique final speed.

<p>The speed-time graph of a motor vehicle is as shown.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1765958805149-f0rl3bk51an.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjU5NTg4MDUxNDktZjBybDNiazUxYW4ucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjc4OTI2MywiZXhwIjoxNzkyNzg5NTYzfQ.AklAWdysOPedDDQPdF2VKOu63FNJg2nIrdTyaXHrsjE" style="max-width: min(450px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the total distance traveled by the motorist.</div></li><li><div class="list-item-paragraph">Find the speed of the motorist when time is $28$ seconds.</div></li><li><div class="list-item-paragraph">Draw the acceleration-time graph for the whole journey.</div></li></ol>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1765958861643-v52romd17bp.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjU5NTg4NjE2NDMtdjUycm9tZDE3YnAucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjc4OTI2MywiZXhwIjoxNzkyNzg5NTYzfQ.dP9NNxYvg5tjMD-_oYuFrqQpdeS7r3qRyhBec332pn4" style="max-width: min(450px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>

The speed-time graph of a motor vehicle is as shown.

Speed-Time Graphs

<p>The diagram below shows the speed-time graph of a particle moving in a straight line.</p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="abc-list"><li><div class="list-item-paragraph">Calculate the acceleration of the particle during the first $15$ seconds.</div></li><li><div class="list-item-paragraph">Calculate the distance moved by the particle in the first $50$ seconds of its motion.</div></li><li><div class="list-item-paragraph">Given that the particle decelerates at $2$ m/s$^{2}$, find the value of $t$.</div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1765958238306-u860j1f5iup.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjU5NTgyMzgzMDYtdTg2MGoxZjVpdXAucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjc4OTI2MywiZXhwIjoxNzkyNzg5NTYzfQ.tZdMACIDjdGUdTCOVQ7Drmdhvb_l4TKuEhkY-ue9fKc" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div></div></div></div>

The diagram below shows the speed-time graph of a particle moving in a straight line.

<p>The diagram shows the speed-time graph of a truck moving in the first $60$ seconds.</p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="abc-list"><li><div class="list-item-paragraph">If the truck accelerates at $0.7$ m/s$^{2}$ in the first $20$ seconds, find the value of $v$.</div></li><li><div class="list-item-paragraph">If the truck travels at constant speed of $18$ m/s for a distance of $252$ m, find the value of $t$.</div></li><li><div class="list-item-paragraph">Find the speed of the truck after $50$ seconds.</div></li><li><div class="list-item-paragraph">Calculate the average speed of the truck for the whole journey.</div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1765957815785-a23y541o208.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjU5NTc4MTU3ODUtYTIzeTU0MW8yMDgucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjc4OTI2MywiZXhwIjoxNzkyNzg5NTYzfQ.rzwYNX9NFUugCTK2FTIpRGZSRAFMKZeHmlywsgxA1-Y" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div></div></div></div>

The diagram shows the speed-time graph of a truck moving in the first $60$ seconds.

<p>The graph below shows the distance-time graphs of a float race between John and Roger.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1765957143698-ur3ieoo4j1b.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjU5NTcxNDM2OTgtdXIzaWVvbzRqMWIucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjc4OTI2MywiZXhwIjoxNzkyNzg5NTYzfQ.vgl4JRJ5NEx9HbxeYKkyi1gPUHSAqW82VVR8akg2uYg" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-paragraph">How long does it take Roger to travel the first $250$m?</div></li><li><div class="list-item-paragraph">What is the distance between John and Roger when&nbsp;$t=40$?</div></li><li><div class="list-item-paragraph">Describe the motion of John’s float from $t=40$ to $t=70$.</div></li><li><div class="list-item-paragraph">Find John’s speed when&nbsp;$t=85$.</div></li><li><div class="list-item-paragraph">What is Roger’s speed when&nbsp;$t=60$?</div></li></ol>

The graph below shows the distance-time graphs of a float race between John and Roger.

Distance-Time Graphs

<p>The distance-time graphs show the journeys of two cyclists $A$ and $B$.<br>Cyclist $A$ travelled from $P$ to $Q$ while cyclist $B$ travelled from $Q$ to $P$.<br>Use the graph to answer each of the following questions.</p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--1-2" data-layout="1-2"><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="abc-list"><li><div class="list-item-paragraph">Find the distance $PQ$.</div></li><li><div class="list-item-paragraph">How long did cyclist $B$ rest during the journey?</div></li><li><div class="list-item-paragraph">At what time did cyclists $A$ and $B$ meet?</div></li><li><div class="list-item-paragraph">Find the average speed of cyclist $B$ for the whole journey.</div></li></ol></div><div class="question-column" style="flex: 2; align-self: flex-start;"><div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1765956551922-tn3vz5fgqcc.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjU5NTY1NTE5MjItdG4zdno1ZmdxY2MucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjc4OTI2MywiZXhwIjoxNzkyNzg5NTYzfQ.ePlImB2Z7Qr4ShLthFeofhRtl7zWXJpQtnqy4ipB-hQ" style="max-width: min(500px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div></div></div></div>

The distance-time graphs show the journeys of two cyclists $A$ and $B$.
Cyclist $A$ travelled from $P$ to $Q$ while cyclist $B$ travelled from $Q$ to $P$.
Use the graph to answer each of the following questions.

<p>Hakim travelled to visit his friend at town $E$. During the journey, he stopped for breakfast at a coffeeshop, after which he stopped at a souvenir shop to buy gift for his friend before continuing his journey to his friend’s town.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1763753656794-zt4eungiit.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjM3NTM2NTY3OTQtenQ0ZXVuZ2lpdC5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyNzg5MjYzLCJleHAiOjE3OTI3ODk1NjN9.36oIZ4bvmG5CxRLI_QfWgstMiJxo6xKbnOCnJzdVBq8" style="max-width: min(450px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li data-list-item-id="item-1763753665733-3wdiv9n7m" data-list-item-path="item-1763753665733-3wdiv9n7m"><div class="list-item-paragraph">How far is it from his home to the souvenir shop? </div></li><li data-list-item-id="item-1763753666294-bexm6s0tw" data-list-item-path="item-1763753666294-bexm6s0tw"><div class="list-item-paragraph">Find the time Hakim spent</div><ol class="i-list"><li data-list-item-id="subitem-1763753687988-uvpf3t5j6" data-list-item-path="item-1763753666294-bexm6s0tw/subitem-1763753687988-uvpf3t5j6"><div class="list-item-paragraph">eating breakfast,</div></li><li data-list-item-id="subitem-1763753689213-dcdsom8j9" data-list-item-path="item-1763753666294-bexm6s0tw/subitem-1763753689213-dcdsom8j9"><div class="list-item-paragraph">at the souvenir shop. </div></li></ol></li><li data-list-item-id="item-1763753667120-zlodenrlf" data-list-item-path="item-1763753667120-zlodenrlf"><div class="list-item-paragraph">Find the gradient of each of the following line segments, stating clearly what each gradient represents. </div><ol class="i-list"><li data-list-item-id="subitem-1763753707448-ngjcpnb81" data-list-item-path="item-1763753667120-zlodenrlf/subitem-1763753707448-ngjcpnb81"><div class="list-item-paragraph">$OA$, </div></li><li data-list-item-id="subitem-1763753708205-zusfzmv7g" data-list-item-path="item-1763753667120-zlodenrlf/subitem-1763753708205-zusfzmv7g"><div class="list-item-paragraph">$BC$, </div></li><li data-list-item-id="subitem-1763753708828-l4flb2s0x" data-list-item-path="item-1763753667120-zlodenrlf/subitem-1763753708828-l4flb2s0x"><div class="list-item-paragraph">$DE$. </div></li></ol></li></ol>

Hakim travelled to visit his friend at town $E$. During the journey, he stopped for breakfast at a coffeeshop, after which he stopped at a souvenir shop to buy gift for his friend before continuing his journey to his friend’s town.

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