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PSLE

PSLE Foundation Mathematics

<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 train leaves a station from rest and travels with uniform acceleration. It attains a maximum speed of \(20\) ms\(^{-1}\) after \(4.8\) km. After which it slows down with a uniform acceleration of \(-0.25\) ms\(^{-2}\) to stop at the next station. Find the distance between the stations and the total time of the journey.</p>
<p>Take the direction towards the next station as positive.</p>

A train leaves a station from rest and travels with uniform acceleration. It attains a maximum speed of \(20\) ms\(^{-1}\) after \(4.8\) km. After which it slows down with a uniform acceleration of \(-0.25\) ms\(^{-2}\) to stop at the next station. Find the distance between the stations and the total time of the journey.

Take the direction towards the next station as positive.

Constant-Acceleration Motion

<p>A stone is thrown vertically upwards from a bridge over a river with a speed of \(8\) ms\(^{-1}\).</p>
<p>It hits the water with a speed of \(30\) ms\(^{-1}\).</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"><p>the height of the bridge over the water,</p></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph"><p>the time taken for the stone to hit the water,</p></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph"><p>the total distance the stone has travelled on its journey.</p></div></li></ol>
<p>Neglect air resistance and use constant gravitational acceleration \(g=9.81\text{ m/s}^2\).</p>

A stone is thrown vertically upwards from a bridge over a river with a speed of \(8\) ms\(^{-1}\).

It hits the water with a speed of \(30\) ms\(^{-1}\).

Neglect air resistance and use constant gravitational acceleration \(g=9.81\text{ m/s}^2\).

<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

Finding the Equation of Displacement from Velocity

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

Differentiating Displacement and Velocity

<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>Two particles, \(A\) and \(B\), leave a point \(O\) at the same time and initially travel in the same direction along the same straight line. Particle \(A\) starts with a velocity \(3 \frac{1}{3} m s^{-1}\) and moves with a constant acceleration of \(4 \frac{1}{3} m s^{-2}\). Particle \(B\) starts with a velocity of \(8 m s^{-1}\) and moves with an acceleration of \(a m s^{-2}\), where&nbsp;\(a = 3 - 2t\) and time \(t\) is the time in seconds after passing point \(O\).</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 velocity of each 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">the displacement of each particle in terms of \(t\).</div></li></ol>
<p>Hence, find</p>
<ol class="abc-list" start="3"><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">the distance travelled by Particle \(B\) in the first \(6\) seconds,</div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">the distance from \(O\) at which \(A\) collides with \(B\).</div></li></ol>

Two particles, \(A\) and \(B\), leave a point \(O\) at the same time and initially travel in the same direction along the same straight line. Particle \(A\) starts with a velocity \(3 \frac{1}{3} m s^{-1}\) and moves with a constant acceleration of \(4 \frac{1}{3} m s^{-2}\). Particle \(B\) starts with a velocity of \(8 m s^{-1}\) and moves with an acceleration of \(a m s^{-2}\), where \(a = 3 - 2t\) and time \(t\) is the time in seconds after passing point \(O\).

the velocity of each particle in terms of \(t\),

the displacement of each particle in terms of \(t\).

the distance travelled by Particle \(B\) in the first \(6\) seconds,

the distance from \(O\) at which \(A\) collides with \(B\).

Comparing One-Dimensional Motions

<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.

Maximum and Minimum Motion Quantities

<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 particle travels in a straight line so that, \(t\) seconds after passing a fixed point \(O\), its displacement, \(s\) metres is given by \(s = t^2 + t\).</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 value of \(s\) when&nbsp;\(t = 3\)</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Find the total distance travelled by the particle during the first \(3\) seconds</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 \(s = t^2 + t\).

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

<p>A particle \(P\) moves in a straight line so that, \(t\) s after passing a fixed point \(O\), its acceleration, \(a\) ms\({}^{-2}\) is given by \(a = -3e^{-\frac{1}{3}t}\). Given that&nbsp;\(P\) has an initial velocity of \(7\) ms\({}^{-1}\), 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">value of \(t\) when&nbsp;\(P\) is instantaneously at rest.</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">total distance travelled by \(P\) between \(t=0\) to \(t=10\).</div></li></ol>

A particle \(P\) moves in a straight line so that, \(t\) s after passing a fixed point \(O\), its acceleration, \(a\) ms\({}^{-2}\) is given by \(a = -3e^{-\frac{1}{3}t}\). Given that \(P\) has an initial velocity of \(7\) ms\({}^{-1}\), find the

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

total distance travelled by \(P\) between \(t=0\) to \(t=10\).

<p>A particle \(P\) leaves a fixed point \(O\) and moves in a straight line so that, \(t\) s after leaving point \(O\), its velocity \(v\) m/s, is given by \(v = 3\sin 2t + 2\).</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find, when \(t = 10\),</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">the velocity of \(P\),</span></div></li><li><div class="list-item-content"><span class="list-item-text">the displacement of \(P\) from \(O\).</span></div></li></ol></li><li><div class="list-item-content"><span class="list-item-text">Calculate</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">the value of \(t\) when \(P\) comes to instantaneous rest for the second time,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the total distance it has travelled when \(t = \frac{3}{4}\pi\).</span></div></li></ol></li></ol>

A particle \(P\) leaves a fixed point \(O\) and moves in a straight line so that, \(t\) s after leaving point \(O\), its velocity \(v\) m/s, is given by \(v = 3\sin 2t + 2\).

<p>A particle \(P\) starts from rest at a fixed-point \(O\) and moves in a straight line. The acceleration, \(a\) ms\({}^{-2}\), of the particle, \(t\) s after passing through \(O\) is given by \(a = 8 - ct^2\), where \(c\) is a positive constant. The velocity of the particle is \(10\) ms\({}^{-1}\) after \(2\) seconds. Find</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">the value of \(c\),</span></div></li><li><div class="list-item-content"><span class="list-item-text">the time at which \(P\) will change its direction of motion,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the average speed of the particle in the first \(5\) seconds.</span></div></li></ol>

A particle \(P\) starts from rest at a fixed-point \(O\) and moves in a straight line. The acceleration, \(a\) ms\({}^{-2}\), of the particle, \(t\) s after passing through \(O\) is given by \(a = 8 - ct^2\), where \(c\) is a positive constant. The velocity of the particle is \(10\) ms\({}^{-1}\) after \(2\) seconds. Find

the time at which \(P\) will change its direction of motion,

the average speed of the particle in the first \(5\) seconds.

<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 \(P\) 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 \(P\) 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

<p>Two particles, \(A\) and \(B\), leave a point \(O\) at the same time and initially travel in the same direction along the same straight line. Particle \(A\) starts with a velocity of \(4\frac{2}{3}\) m/s and moves with a constant acceleration of \(5\frac{1}{3}\) m/\(\mathrm{s}^2\). Particle \(B\) starts with a velocity of \(10\) m/s and moves with an acceleration of \(a\) m/\(\mathrm{s}^2\), where&nbsp;\(a = 8 - 4t\) and \(t\) is the time in seconds after passing point \(O\).</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 velocity of each 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">the displacement of each particle in terms of \(t\).</div></li></ol>
<p>Hence, find</p>
<ol class="abc-list" start="3"><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">the distance travelled by Particle \(B\) in the first \(6\) seconds,</div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">the distance from \(O\) at which \(A\) collides with \(B\).</div></li></ol>

Two particles, \(A\) and \(B\), leave a point \(O\) at the same time and initially travel in the same direction along the same straight line. Particle \(A\) starts with a velocity of \(4\frac{2}{3}\) m/s and moves with a constant acceleration of \(5\frac{1}{3}\) m/\(\mathrm{s}^2\). Particle \(B\) starts with a velocity of \(10\) m/s and moves with an acceleration of \(a\) m/\(\mathrm{s}^2\), where \(a = 8 - 4t\) and \(t\) is the time in seconds after passing point \(O\).

<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 its displacement, \(s\) meters, from a fixed point \(O\) on the straight line is given by \(s=-t^3+3t^2+9t-5\) for \(t \le 4\), where \(t\) is the time in seconds after passing through a point \(P\) on the line.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">the distance \(OP\),</span></div></li><li><div class="list-item-content"><span class="list-item-text">the distance from \(O\) of the particle when it 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 during the first \(4\) seconds,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the velocity of the particle when \(t=4\).</span></div></li></ol></li><li><div class="list-item-content"><span class="list-item-text">At \(t=4\) the acceleration of the particle is changed to \((2t-10)\) \(\mathrm{ms}^{-2}\) the instantaneous velocity remaining unchanged and it finally stops at \(Q\) when \(t=T\). Find the value of \(T\).</span></div></li></ol>

A particle moves in a straight line so that its displacement, \(s\) meters, from a fixed point \(O\) on the straight line is given by \(s=-t^3+3t^2+9t-5\) for \(t \le 4\), where \(t\) is the time in seconds after passing through a point \(P\) on the line.

At \(t=4\) the acceleration of the particle is changed to \((2t-10)\) \(\mathrm{ms}^{-2}\) the instantaneous velocity remaining unchanged and it finally stops at \(Q\) when \(t=T\). Find the value of \(T\).

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