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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 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

Maximum and Minimum Motion Quantities

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 \(A\).

<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 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>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>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 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>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 \(P\) 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>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 \(T\).

Differentiating Displacement and Velocity

<p>A particle, travelling in a straight line, passes a fixed-point \(O\) on the line with a velocity of \(2\) \(\mathrm{ms}^{-1}\). The acceleration, \(a\) \(\mathrm{ms}^{-2}\), of the particle, \(t\) s after passing \(O\), is given by \(a = -4e^{-t}\).</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Show that the particle comes instantaneously to rest when \(t = -\ln \frac{1}{2}\).</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the total distance travelled by the particle between \(t = 0\) and \(t = 2\).</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the average speed of the particle during the first \(2\) seconds.</span></div></li></ol>

A particle, travelling in a straight line, passes a fixed-point \(O\) on the line with a velocity of \(2\) \(\mathrm{ms}^{-1}\). The acceleration, \(a\) \(\mathrm{ms}^{-2}\), of the particle, \(t\) s after passing \(O\), is given by \(a = -4e^{-t}\).

Show that the particle comes instantaneously to rest when \(t = -\ln \frac{1}{2}\).

Find the total distance travelled by the particle between \(t = 0\) and \(t = 2\).

Find the average speed of the particle during the first \(2\) seconds.

Math Question Bank