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PSLE Foundation Mathematics

<p>A particle \(P\) moves in a straight line from a fixed point \(O\). Its displacement, \(s\) metres, from \(O\) at time \(t\) seconds is given by \(s=2t^3-9t^2+12t\), where&nbsp;\(t\geq0\).</p>
<ol class="i-list"><li data-list-item-id="amkin-q01-parts-item-1" data-list-item-path="amkin-q01-parts-item-1"><div class="list-item-paragraph">Find the displacement of \(P\) when&nbsp;\(t=3\).<span class="inline-marks">[1]</span></div></li><li data-list-item-id="amkin-q01-parts-item-2" data-list-item-path="amkin-q01-parts-item-2"><div class="list-item-paragraph">Find the positive values of \(t\) for which \(P\) is instantaneously at rest.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="amkin-q01-parts-item-3" data-list-item-path="amkin-q01-parts-item-3"><div class="list-item-paragraph">Find the range of values of \(t\) for which the acceleration of \(P\) is positive.<span class="inline-marks">[2]</span></div></li></ol>

A particle \(P\) moves in a straight line from a fixed point \(O\). Its displacement, \(s\) metres, from \(O\) at time \(t\) seconds is given by \(s=2t^3-9t^2+12t\), where \(t\geq0\).

Find the displacement of \(P\) when \(t=3\).

Find the positive values of \(t\) for which \(P\) is instantaneously at rest.

Find the range of values of \(t\) for which the acceleration of \(P\) is positive.

Differentiating Displacement and Velocity

Motion Conditions, Rest and Direction

Cubic displacement and acceleration sign

Kinematics

<p>A particle travels in a straight line so that its velocity, \(v\) m/s, is given by \(v = -2t^2 + 6t + 4\), where&nbsp;\(t\) seconds is the time after passing \(O\). Find an expression in terms of \(t\) for</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph"><p>its acceleration,</p></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph"><p>its displacement from \(O\),</p></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph"><p>the values of \(t\) when the speed is \(3\) m/s,</p></div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph"><p>the values of \(t\) when the displacement of the particle is \(0\) m.</p></div></li></ol>

A particle travels in a straight line so that its velocity, \(v\) m/s, is given by \(v = -2t^2 + 6t + 4\), where \(t\) seconds is the time after passing \(O\). Find an expression in terms of \(t\) for

Finding the Equation of Displacement from Velocity

<p>A particle moves in a straight line with a velocity, \(v\) m s\(^{-1}\). Its velocity can be modelled by the equation \(v = \sin t + t \cos t\). Given the initial conditions \(t = 0\), \(s = 0\).</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Differentiate \(s = t \sin t\) with respect to \(t\).</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Hence find an expression for the displacement, \(s\) at time \(t\)</div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Find an expression for the acceleration, \(a\), at time \(t\).</div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">Determine whether the motion is periodic, and justify your answer.</div></li><li data-list-item-id="item-e" data-list-item-path="item-e"><div class="list-item-paragraph">What happens to the amplitude of the motion as time progresses?</div></li></ol>

A particle moves in a straight line with a velocity, \(v\) m s\(^{-1}\). Its velocity can be modelled by the equation \(v = \sin t + t \cos t\). Given the initial conditions \(t = 0\), \(s = 0\).

Differentiate \(s = t \sin t\) with respect to \(t\).

Hence find an expression for the displacement, \(s\) at time \(t\)

Find an expression for the acceleration, \(a\), at time \(t\).

Determine whether the motion is periodic, and justify your answer.

What happens to the amplitude of the motion as time progresses?

Integration as an Inverse Process of Differentiation

Interpreting Motion Models

Product Rule

Integration

Differentiation

<p>A particle travels in a straight line such that, \(t\) s after passing a fixed point, \(O\), its velocity is given by \(v = 8 \cos \left( \frac{t}{2} \right).\) Find:</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">The value of \(t\) when&nbsp;\(v = 4\) for the first time</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">The acceleration when&nbsp;\(t = 2\)</div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">The distance from \(O\) when&nbsp;\(v = 0\) for the first time.</div></li></ol>

A particle travels in a straight line such that, \(t\) s after passing a fixed point, \(O\), its velocity is given by \(v = 8 \cos \left( \frac{t}{2} \right).\) Find:

The value of \(t\) when \(v = 4\) for the first time

The distance from \(O\) when \(v = 0\) for the first time.

<p>A particle moving in a straight line has a displacement, \(t\) seconds after passing a fixed point \(O\), given by the formula \(s = te^{-0.2t}\). Find:</p>
<ol class="abc-list"><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">The velocity, in \(m/s\), of the particle when&nbsp;\(t = 4\)</div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">The acceleration of the particle when&nbsp;\(t = 2\)</div></li></ol>

A particle moving in a straight line has a displacement, \(t\) seconds after passing a fixed point \(O\), given by the formula \(s = te^{-0.2t}\). Find:

The velocity, in \(m/s\), of the particle when \(t = 4\)

The acceleration of the particle when \(t = 2\)

<p>A particle moving in a straight line has displacement \(s\) metres from a fixed point \(O\), at time \(t\) seconds after the motion begins, given by the formula \(s = 2 \cos \left( \frac{\pi t}{4} \right)\). Find:</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">The velocity, in m/s, of the particle when&nbsp;\(t=4\)</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">The acceleration of the particle when&nbsp;\(t=2\)</div></li></ol>

A particle moving in a straight line has displacement \(s\) metres from a fixed point \(O\), at time \(t\) seconds after the motion begins, given by the formula \(s = 2 \cos \left( \frac{\pi t}{4} \right)\). Find:

The velocity, in m/s, of the particle when \(t=4\)

The acceleration of the particle when \(t=2\)

<p>A particle moves in a straight line from a point \(O\), such that&nbsp;\(t\) seconds after leaving \(O\), its velocity, \(v\) m/s is given by \(v = e^{0.2t} + 4\).</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Calculate the initial acceleration of the particle.</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Calculate the displacement of the particle when&nbsp;\(t = 2\) s</div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Will the particle return to \(O\)?</div></li></ol>

A particle moves in a straight line from a point \(O\), such that \(t\) seconds after leaving \(O\), its velocity, \(v\) m/s is given by \(v = e^{0.2t} + 4\).

Calculate the initial acceleration of the particle.

Calculate the displacement of the particle when \(t = 2\) s

<p>A particle \(P\) moves in a straight line so that its displacement, \(s\) metres, from a fixed point \(O\) is given by \(s = 12t^2 - t^4\), where&nbsp;\(t\) seconds is the amount of time after leaving \(O\). Find</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">The value of \(s\) where the particle \(P\) is instantaneously at rest apart from the start point</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">The distance travelled by \(P\) after \(2\) seconds</div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">The velocity of \(P\) when the acceleration is \(-0.05ms^{-2}\)</div></li></ol>

A particle \(P\) moves in a straight line so that its displacement, \(s\) metres, from a fixed point \(O\) is given by \(s = 12t^2 - t^4\), where \(t\) seconds is the amount of time after leaving \(O\). Find

The value of \(s\) where the particle \(P\) is instantaneously at rest apart from the start point

The distance travelled by \(P\) after \(2\) seconds

The velocity of \(P\) when the acceleration is \(-0.05ms^{-2}\)

Distance Travelled and Average Speed

<p>A particle moves in a straight line and its displacement, \(s\) metres, from a fixed point \(O\) is given by the equation \(s = 24t - t^3\). Calculate</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">The acceleration of the particle when it comes instantaneously to rest.</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">The velocity of the particle when it is at \(O\) again</div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">The distance travelled by the particle during the first \(3\) seconds</div></li></ol>
<p>Here \(t\ge0\) is the time in seconds from the start of the motion.</p>

A particle moves in a straight line and its displacement, \(s\) metres, from a fixed point \(O\) is given by the equation \(s = 24t - t^3\). Calculate

The acceleration of the particle when it comes instantaneously to rest.

The velocity of the particle when it is at \(O\) again

The distance travelled by the particle during the first \(3\) seconds

Here \(t\ge0\) is the time in seconds from the start of the motion.

<p>The height, \(h\) m, of a stone \(t\) seconds after it has been thrown vertically upwards from ground level is given by \(h = 36t - 6t^{2}\). Find</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph"><p>its velocity after \(3\) seconds,</p></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph"><p>the maximum height reached,</p></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph"><p>the time of flight.</p></div></li></ol>

The height, \(h\) m, of a stone \(t\) seconds after it has been thrown vertically upwards from ground level is given by \(h = 36t - 6t^{2}\). Find

Maximum and Minimum Motion Quantities

<p>A particle moves in a straight line so that its displacement, \(s\) metres, from a fixed point \(O\) on the straight line is given by \(s = -t^3 + 2t^2 + 8t - 6\) for&nbsp;\(t \le 3\), where&nbsp;\(t\) is the time in seconds after passing through a point \(P\) on the line.</p>
<p>Find</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">The distance \(OP\)</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">The distance from \(O\) of the particle when it is instantaneously at rest,</div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">The total distance travelled by the particle during the first \(3\) seconds</div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">The velocity of the particle when&nbsp;\(t = 3\)</div></li></ol>

A particle moves in a straight line so that its displacement, \(s\) metres, from a fixed point \(O\) on the straight line is given by \(s = -t^3 + 2t^2 + 8t - 6\) for \(t \le 3\), where \(t\) is the time in seconds after passing through a point \(P\) on the line.

The distance from \(O\) of the particle when it is instantaneously at rest,

The total distance travelled by the particle during the first \(3\) seconds

The velocity of the particle when \(t = 3\)

<p>A particle moves in a straight line with a velocity \(v\) \(m s^{-1}\) given by \(v = 3t^2 - 4t - 8\), where&nbsp;\(t\) is the time in seconds after passing a point \(A\), which is \(4\) metres to the left of a fixed point \(O\).</p>
<p>Find the</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">acceleration when&nbsp;\(t=4\).</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">The minimum velocity.</div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">The average speed during the first \(4\) seconds.</div></li></ol>

A particle moves in a straight line with a velocity \(v\) \(m s^{-1}\) given by \(v = 3t^2 - 4t - 8\), where \(t\) is the time in seconds after passing a point \(A\), which is \(4\) metres to the left of a fixed point \(O\).

The average speed during the first \(4\) seconds.

<p>A particle \(P\) moves in a straight line so that its displacement, \(s\) metres, from a fixed point \(O\), is given by \(s = t^3 - 2t^2 - 12t - 5\), where time \(t\) is the time taken in seconds after the start of motion. Calculate</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">the initial velocity and acceleration,</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">the value(s) of \(t\) when&nbsp;\(P\) is instantaneously at rest,</div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">the distance travelled in the first \(5\) seconds,</div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">the average speed of \(P\) during the first \(5\) seconds</div></li></ol>

A particle \(P\) moves in a straight line so that its displacement, \(s\) metres, from a fixed point \(O\), is given by \(s = t^3 - 2t^2 - 12t - 5\), where time \(t\) is the time taken in seconds after the start of motion. Calculate

the value(s) of \(t\) when \(P\) is instantaneously at rest,

the distance travelled in the first \(5\) seconds,

the average speed of \(P\) during the first \(5\) seconds

<p>A polar bear moves in a straight line to a food source and its path can be modelled by the equation \(s = 10t - t^2\) where&nbsp;\(s\) m is the distance from the nest at time \(t\) s.</p>
<p>The polar bear collects the food and then returns to its original location.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Find an expression for the velocity of the bear.</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Find how far away the source of food is.</div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Find the length of time of the bear's journey.</div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">What is the main problem with this model?</div></li></ol>
<p>Use the model from departure at \(t=0\) until the first return to the nest. It is not a nonnegative distance-from-nest model beyond that journey.</p>

A polar bear moves in a straight line to a food source and its path can be modelled by the equation \(s = 10t - t^2\) where \(s\) m is the distance from the nest at time \(t\) s.

The polar bear collects the food and then returns to its original location.

Find an expression for the velocity of the bear.

Find the length of time of the bear's journey.

What is the main problem with this model?

Use the model from departure at \(t=0\) until the first return to the nest. It is not a nonnegative distance-from-nest model beyond that journey.

<p>A particle, \(P\), moves in a straight line. Its displacement, \(x\) m, from a fixed point \(O\) at time \(t\) is given by \(x = t(t^2 - 16)\). <br>Find</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph"><p>the times when the particle is at \(O\),</p></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph"><p>an expression for the velocity of the particle,</p></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph"><p>when the particle is at rest,</p></div></li></ol>
<p>Here \(t\ge0\) is the time in seconds from the start of the motion.</p>

A particle, \(P\), moves in a straight line. Its displacement, \(x\) m, from a fixed point \(O\) at time \(t\) is given by \(x = t(t^2 - 16)\). 
Find

<p>A particle travels in a straight line so that, \(t\) seconds after passing a fixed point \(O\), its displacement, \(s\) metres is given by \(s = 9t^2 - 2t^3\).</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Find an expression for the velocity, \(v\) m/s, of the particle in terms of \(t\).</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Find an expression for the acceleration, \(a\) m/s\(^2\), of the particle at any time \(t\) seconds</div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Find the acceleration at \(t = 2\)</div></li></ol>

A particle travels in a straight line so that, \(t\) seconds after passing a fixed point \(O\), its displacement, \(s\) metres is given by \(s = 9t^2 - 2t^3\).

Find an expression for the velocity, \(v\) m/s, of the particle in terms of \(t\).

Find an expression for the acceleration, \(a\) m/s\(^2\), of the particle at any time \(t\) seconds

<p>A particle travels in a straight line such that, \(t\) seconds after passing a fixed point \(O\), its displacement, \(s\) metres is given by \(s = 13t^{2} - 8t\).</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Find an expression for the velocity \(v\) m/s,</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Find the velocity at \(t = 5\)s</div></li></ol>

A particle travels in a straight line such that, \(t\) seconds after passing a fixed point \(O\), its displacement, \(s\) metres is given by \(s = 13t^{2} - 8t\).

Find an expression for the velocity \(v\) m/s,

<p>A particle \(P\) moves along a straight line such that&nbsp;\(t\) seconds after passing through a fixed point \(O\), its velocity is given by \(v = \frac{1}{2}e^{2t} - 4\). The particle comes to instantaneous rest at a point \(A\).</p>
<ol class="abc-list"><li data-list-item-id="item-q5-a" data-list-item-path="item-q5-a"><div class="list-item-paragraph">Find an expression, in terms of \(t\), for the acceleration of \(P\).</div></li><li data-list-item-id="item-q5-b" data-list-item-path="item-q5-b"><div class="list-item-paragraph">Find an expression, in terms of \(t\), for the displacement of \(P\).</div></li><li data-list-item-id="item-q5-c" data-list-item-path="item-q5-c"><div class="list-item-paragraph">Show that the time taken for&nbsp;\(P\) to travel from \(O\) to \(A\) is \(\frac{1}{2}\ln 8\) seconds.</div></li><li data-list-item-id="item-q5-d" data-list-item-path="item-q5-d"><div class="list-item-paragraph">Given that&nbsp;\(P\) passes through a point \(B\) when&nbsp;\(t = 3\), find the average speed of \(P\) from \(A\) to \(B\).</div></li></ol>

A particle \(P\) moves along a straight line such that \(t\) seconds after passing through a fixed point \(O\), its velocity is given by \(v = \frac{1}{2}e^{2t} - 4\). The particle comes to instantaneous rest at a point \(A\).

Find an expression, in terms of \(t\), for the acceleration of \(P\).

Find an expression, in terms of \(t\), for the displacement of \(P\).

Show that the time taken for \(P\) to travel from \(O\) to \(A\) is \(\frac{1}{2}\ln 8\) seconds.

Given that \(P\) passes through a point \(B\) when \(t = 3\), find the average speed of \(P\) from \(A\) to \(B\).

<p>A particle moves in a straight line so that, at time \(t\) seconds after leaving a fixed point \(O\), its velocity, \(v\) ms\({}^{-1}\), is given by \(v = 7 - 6e^{1-\frac{1}{2}t}\). Find</p>
<ol class="abc-list"><li data-list-item-id="item-q4-a" data-list-item-path="item-q4-a"><div class="list-item-paragraph">the initial acceleration of the particle,</div></li><li data-list-item-id="item-q4-b" data-list-item-path="item-q4-b"><div class="list-item-paragraph">the value of \(t\) when the particle is instantaneously at rest,</div></li><li data-list-item-id="item-q4-c" data-list-item-path="item-q4-c"><div class="list-item-paragraph">the distance of the particle from \(O\) when&nbsp;\(t = 2\).</div></li></ol>

A particle moves in a straight line so that, at time \(t\) seconds after leaving a fixed point \(O\), its velocity, \(v\) ms\({}^{-1}\), is given by \(v = 7 - 6e^{1-\frac{1}{2}t}\). Find

the initial acceleration of the particle,

the value of \(t\) when the particle is instantaneously at rest,

the distance of the particle from \(O\) when \(t = 2\).

<p>A particle \(P\) travels in a straight line so that its displacement, \(s\) cm, from a fixed point \(O\) is given by \(s = 3t + \frac{12}{t+1}\), where&nbsp;\(t\) is the time in seconds after leaving the starting point \(A\). Calculate</p>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">the initial velocity of \(P\),</div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">the velocity of \(P\) when it is next at \(A\),</div></li><li data-list-item-id="item-iii" data-list-item-path="item-iii"><div class="list-item-paragraph">the minimum displacement of \(P\) from \(O\),</div></li><li data-list-item-id="item-iv" data-list-item-path="item-iv"><div class="list-item-paragraph">the total distance travelled during the first \(3\) seconds.</div></li></ol>

A particle \(P\) travels in a straight line so that its displacement, \(s\) cm, from a fixed point \(O\) is given by \(s = 3t + \frac{12}{t+1}\), where \(t\) is the time in seconds after leaving the starting point \(A\). Calculate

the velocity of \(P\) when it is next at \(A\),

the minimum displacement of \(P\) from \(O\),

the total distance travelled during the first \(3\) seconds.

Math Question Bank