A Math

A Math (IGCSE)

E Math

E Math (IGCSE)

H1 Math

H2 Math

IB AA HL Math

IB AA SL Math

IB AI HL Math

IB AI SL Math

IP Math

N(A) A Math

N(A) E Math

PSLE

PSLE Foundation Mathematics

<p>When a particle has reached an interior local maximum of differentiable displacement with reference to its start point \(O\), what is the velocity of the particle?</p>

When a particle has reached an interior local maximum of differentiable displacement with reference to its start point \(O\), what is the velocity of the particle?

Maximum and Minimum Motion Quantities

Kinematics

<p>At an interior local maximum of a particle’s positive speed, assuming its velocity is differentiable, what is the acceleration of the particle?</p>
<p>Hint: Acceleration is the rate of change of velocity with respect to time, i.e. \(a = \frac{\mathrm{d}v}{\mathrm{d}t}\)</p>

At an interior local maximum of a particle’s positive speed, assuming its velocity is differentiable, what is the acceleration of the particle?

Hint: Acceleration is the rate of change of velocity with respect to time, i.e. \(a = \frac{\mathrm{d}v}{\mathrm{d}t}\)

<p>A particle moving in a straight line passes a fixed point \(O\) on the line with its velocity \(36 \text{ ms}^{-1}\). The acceleration, \(a \text{ ms}^{-2}\), of the particle, \(t\) seconds after passing \(O\), is given by \(a = 5 - 2t\). 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 velocity of the particle \(4\) seconds after passing \(O\)</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">The time taken to reach the maximum distance from \(O\) in the direction of the initial motion</div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">This maximum distance</div></li></ol>

A particle moving in a straight line passes a fixed point \(O\) on the line with its velocity \(36 \text{ ms}^{-1}\). The acceleration, \(a \text{ ms}^{-2}\), of the particle, \(t\) seconds after passing \(O\), is given by \(a = 5 - 2t\). Calculate

The velocity of the particle \(4\) seconds after passing \(O\)

The time taken to reach the maximum distance from \(O\) in the direction of the initial motion

Velocity from Acceleration

Finding the Equation of Displacement from Velocity

<p>A particle starts from rest at \(O\) and moves in a straight line with its acceleration, \(a\,ms^{-2}\), given by the equation \(a = 6 - 3t\), where&nbsp;\(t\) seconds is the time after leaving a fixed point \(O\). The particle comes to instantaneous rest at a point \(A\). 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 time taken for the particle to reach \(A\),</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">The distance \(OA\)</div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">The maximum speed of the particle during its motion from \(O\) to \(A\).</div></li></ol>

A particle starts from rest at \(O\) and moves in a straight line with its acceleration, \(a\,ms^{-2}\), given by the equation \(a = 6 - 3t\), where \(t\) seconds is the time after leaving a fixed point \(O\). The particle comes to instantaneous rest at a point \(A\). Find

the time taken for the particle to reach \(A\),

The maximum speed of the particle during its motion from \(O\) to \(A\).

Motion Conditions, Rest and Direction

<p>A particle travelling in a straight line starts from rest at point \(O\) and has acceleration (in \(\text{m s}^{-2}\)) <br>Modelled by the equation \(a = 12 - 8t\), where&nbsp;\(t\) is measured in seconds.</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 particle at time \(t\).</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 position of the particle at time \(t\).</div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Find its furthest distance from \(O\) in a positive direction during the subsequent motion.</div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">Find the time taken for the particle to return to \(O\), and its velocity at that time.</div></li><li data-list-item-id="item-e" data-list-item-path="item-e"><div class="list-item-paragraph">Describe the motion of the particle after it passes through \(O\) for the second time.</div></li></ol>

A particle travelling in a straight line starts from rest at point \(O\) and has acceleration (in \(\text{m s}^{-2}\)) 
Modelled by the equation \(a = 12 - 8t\), where \(t\) is measured in seconds.

Find an expression for the velocity of the particle at time \(t\).

Find an expression for the position of the particle at time \(t\).

Find its furthest distance from \(O\) in a positive direction during the subsequent motion.

Find the time taken for the particle to return to \(O\), and its velocity at that time.

Describe the motion of the particle after it passes through \(O\) for the second time.

Speeding Up and Slowing Down

<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

Differentiating Displacement and Velocity

<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.

Distance Travelled and Average Speed

<p>The velocity of a particle, in \(ms^{-1}\), moving in a straight line, \(t\) \(s\) after passing a fixed point, \(O\), is given by the equation \(v = 12t - 2t^{2}\). Find:</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">The time at which the velocity reaches a maximum.</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">The positive time at which the particle next comes instantaneously to rest.</div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">The distance from \(O\) at that positive rest time.</div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">The speed of the particle when it passes \(O\) again.</div></li></ol>

The velocity of a particle, in \(ms^{-1}\), moving in a straight line, \(t\) \(s\) after passing a fixed point, \(O\), is given by the equation \(v = 12t - 2t^{2}\). Find:

The time at which the velocity reaches a maximum.

The positive time at which the particle next comes instantaneously to rest.

The distance from \(O\) at that positive rest time.

The speed of the particle when it passes \(O\) again.

<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.

Interpreting Motion Models

<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.

<p>A particle moving in a straight line passes through a fixed point \(O\) with velocity \(2\) \(\mathrm{ms}^{-1}\). The acceleration, \(a\) \(\mathrm{ms}^{-2}\), of the particle \(t\) seconds after passing \(O\) is given by \(a = 5 - 6t\). Calculate</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">the maximum velocity attained by the particle,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the value of \(t\) for which the particle is instantaneously at rest,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the total distance travelled by the particle between \(t = 0\) and \(t = 3\).</span></div></li></ol>

A particle moving in a straight line passes through a fixed point \(O\) with velocity \(2\) \(\mathrm{ms}^{-1}\). The acceleration, \(a\) \(\mathrm{ms}^{-2}\), of the particle \(t\) seconds after passing \(O\) is given by \(a = 5 - 6t\). Calculate

the maximum velocity attained by the particle,

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

the total distance travelled by the particle between \(t = 0\) and \(t = 3\).

<ol class="abc-list"></ol>
<p>A particle moves in a straight line so that, \(t\) seconds after passing through a fixed point \(O\), its velocity \(v\) ms\({}^{-1}\), is given by \(v=3e^t-40e^{-2t}\). Find</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">the initial velocity of the particle.</span></div></li><li><div class="list-item-content"><span class="list-item-text">the value of \(t\) when the particle is instantaneously at rest.</span></div></li><li><div class="list-item-content"><span class="list-item-text">the acceleration of the particle when&nbsp;\(t=\ln 2\).</span></div></li><li><div class="list-item-content"><span class="list-item-text">the value of \(t\) when the acceleration is minimum.</span></div></li><li><div class="list-item-content"><span class="list-item-text">an expression for the displacement in terms of \(t\).</span></div></li></ol>

A particle moves in a straight line so that, \(t\) seconds after passing through a fixed point \(O\), its velocity \(v\) ms\({}^{-1}\), is given by \(v=3e^t-40e^{-2t}\). Find

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

the acceleration of the particle when \(t=\ln 2\).

the value of \(t\) when the acceleration is minimum.

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

<p>A particle moves in a straight line with a velocity \(v\) ms\({}^{-1}\) given by \(v = 3t^2 - 2t - 5\), where \(t\) is the time in seconds after passing a point \(A\), which is \(3\) metres to the left of a fixed point \(O\). </p>
<p>Find the</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">acceleration when \(t = 5\).</span></div></li><li><div class="list-item-content"><span class="list-item-text">the minimum velocity.</span></div></li><li><div class="list-item-content"><span class="list-item-text">the average speed during the first \(2\) seconds.</span></div></li></ol>

A particle moves in a straight line with a velocity \(v\) ms\({}^{-1}\) given by \(v = 3t^2 - 2t - 5\), where \(t\) is the time in seconds after passing a point \(A\), which is \(3\) metres to the left of a fixed point \(O\). 

the average speed during the first \(2\) seconds.

<p>A particle \(P\) leaves a fixed point \(O\) and moves in a straight line so that, \(t\) s after leaving \(O\), its velocity is given by \(v = -3t^2 + kt + 15\), where \(k\) is a constant.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Given that its velocity is \(11\) ms\({}^{-1}\) when \(t = 2\), find the value of \(k\).</div></li><li><div class="list-item-content"><span class="list-item-text">Find the value of \(t\) for which the particle is at instantaneous rest.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the maximum velocity of the particle.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Calculate the total distance travelled by the particle in the first \(4\) seconds.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the time the particle returns to point \(O\).</span></div></li></ol>

A particle \(P\) leaves a fixed point \(O\) and moves in a straight line so that, \(t\) s after leaving \(O\), its velocity is given by \(v = -3t^2 + kt + 15\), where \(k\) is a constant.

Find the value of \(t\) for which the particle is at instantaneous rest.

Find the maximum velocity of the particle.

Calculate the total distance travelled by the particle in the first \(4\) seconds.

Find the time the particle returns to point \(O\).

Math Question Bank