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<p>A particle starts from a fixed point \(O\) with velocity \(4\text{ m s}^{-1}\). For the first \(3\) seconds its acceleration is \(2t\text{ m s}^{-2}\). It then begins a second stage of motion; \(T\) seconds after this stage begins, its acceleration is constant at \(-5\text{ m s}^{-2}\).</p>
<ol class="i-list"><li data-list-item-id="amkin-q23-parts-item-1" data-list-item-path="amkin-q23-parts-item-1"><div class="list-item-paragraph">Find the velocity at the end of the first stage and the distance travelled during that stage.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="amkin-q23-parts-item-2" data-list-item-path="amkin-q23-parts-item-2"><div class="list-item-paragraph">Find the time \(T\) at which the particle first comes to rest.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="amkin-q23-parts-item-3" data-list-item-path="amkin-q23-parts-item-3"><div class="list-item-paragraph">Find the total distance travelled from \(O\) before it first comes to rest.<span class="inline-marks">[3]</span></div></li></ol>

A particle starts from a fixed point \(O\) with velocity \(4\text{ m s}^{-1}\). For the first \(3\) seconds its acceleration is \(2t\text{ m s}^{-2}\). It then begins a second stage of motion; \(T\) seconds after this stage begins, its acceleration is constant at \(-5\text{ m s}^{-2}\).

Find the velocity at the end of the first stage and the distance travelled during that stage.

Find the time \(T\) at which the particle first comes to rest.

Find the total distance travelled from \(O\) before it first comes to rest.

Velocity from Acceleration

Piecewise Motion and Changing Time Origins

Distance Travelled and Average Speed

Motion Conditions, Rest and Direction

Two-stage acceleration and braking

Kinematics

<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

Cubic displacement and acceleration sign

<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 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 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 \(A\).

Maximum and Minimum Motion Quantities

<p>A particle passes a fixed point \(O\) with a velocity of \(5\) m/s and moves in a straight line with an acceleration of \(a = 2(5 - t)\) \(ms^{-2}\), where&nbsp;\(t\) seconds is the time after passing \(O\). 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 value of \(t\) when the velocity is again \(5\) m/s</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">the displacement of the particle at this instant.</div></li></ol>

A particle passes a fixed point \(O\) with a velocity of \(5\) m/s and moves in a straight line with an acceleration of \(a = 2(5 - t)\) \(ms^{-2}\), where \(t\) seconds is the time after passing \(O\). Calculate

the value of \(t\) when the velocity is again \(5\) m/s

the displacement of the particle at this instant.

<p>A particle moves in a straight line. Its velocity \(v\) m/s at time \(t\) seconds is given by \(v = t^2 - 5t - 6\), The displacement \(s\) metres, at time \(t\) seconds is measured from a fixed origin \(O\).</p>
<p>When&nbsp;\(t = 0\), its displacement, \(s=5\) metres, 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 the particle is at rest and the displacement at this instant,</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">the displacement when&nbsp;\(t = 2\) and the total distance travelled by the particle in the first \(2\) seconds</div></li></ol>

A particle moves in a straight line. Its velocity \(v\) m/s at time \(t\) seconds is given by \(v = t^2 - 5t - 6\), The displacement \(s\) metres, at time \(t\) seconds is measured from a fixed origin \(O\).

When \(t = 0\), its displacement, \(s=5\) metres, find

the value of \(t\) when the particle is at rest and the displacement at this instant,

the displacement when \(t = 2\) and the total distance travelled by the particle in the first \(2\) seconds

<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 \(-0.05ms^{-2}\)

<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>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

<p>A particle passes a fixed point \(O\) with a velocity of \(6\) \(ms^{-1}\) and moves in a straight line with an acceleration of \(3(4 - t)\) \(ms^{-2}\), where&nbsp;\(t\) seconds is the time after passing \(O\). 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 value of \(t\) when the velocity is again \(6\) \(ms^{-1}\)</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">The position of the particle at this instant</div></li></ol>

A particle passes a fixed point \(O\) with a velocity of \(6\) \(ms^{-1}\) and moves in a straight line with an acceleration of \(3(4 - t)\) \(ms^{-2}\), where \(t\) seconds is the time after passing \(O\). Calculate

The value of \(t\) when the velocity is again \(6\) \(ms^{-1}\)

The position of the particle at this instant

<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 \(P\) 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>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, \(P\), 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 acceleration, \(a\) \(\text{m/s}^2\), is given by \(a = t - 7\).</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Find the expression for&nbsp;\(v\), given that the initial velocity is \(-15\) \(\text{m/s}\).</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Find the value of \(t\) for which the particle is instantaneously at rest.</div></li></ol>

A particle travels in a straight line so that, \(t\) seconds after passing a fixed point \(O\), its acceleration, \(a\) \(\text{m/s}^2\), is given by \(a = t - 7\).

Find the expression for \(v\), given that the initial velocity is \(-15\) \(\text{m/s}\).

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

<p>The acceleration \(a\) \(m s^{-2}\), of a moving particle is given by \(a = 2t - 3t^2\) where&nbsp;\(t\) is the time in seconds. Find the value of \(t\) when the particle</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph"><p>has acceleration \(0.05\) m s\(^{-2}\),</p></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph"><p>has acceleration \(-3\) m s\(^{-2}\).</p></div></li></ol>

The acceleration \(a\) \(m s^{-2}\), of a moving particle is given by \(a = 2t - 3t^2\) where \(t\) is the time in seconds. Find the value of \(t\) when the particle

Math Question Bank