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<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.

Given the Displacement Function

Cubic displacement and acceleration sign

Kinematics

<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 of the particle and the distance travelled during the first 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 of the particle and the distance travelled during the first 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.

Given the Acceleration Function

Two-stage acceleration and braking

<p>Let&nbsp;\(g(x) = x^3 - 3x^2 -9x +5\).</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the two values of \(x\) at which the tangent to the graph of \(g\) is horizontal.<span class="inline-marks">[8]</span></div></li><li><div class="list-item-paragraph">For each of these values, determine whether it is a maximum or a minimum.<span class="inline-marks">[6]</span></div></li></ol>

<p>An object was launched from a machine. Its height, \(y\) metres above ground, \(t\) seconds after launch, can be modelled by a quadratic function \(y = f(t)\). Four seconds after launch, the object reaches its maximum height of \(81\) m and then dropped to a height of \(54\) m after a further \(3\) seconds. </p>
<ol class="abc-list"><li data-list-item-id="item-1773454586595-nxo7pguih"><div class="list-item-paragraph">Find the initial height of the object when it was launched.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1773454587503-2ajg6lta7"><div class="list-item-paragraph">Without solving a quadratic equation, explain and determine how long did the object stay above \(54\) m ?<span class="inline-marks">[2]</span></div></li></ol>

An object was launched from a machine. Its height, \(y\) metres above ground, \(t\) seconds after launch, can be modelled by a quadratic function \(y=f(t)\). Four seconds after launch, the object reached its maximum height of 81 m and then dropped to a height of 54 m after a further three seconds.

Find the initial height of the object when it was launched.

Without solving a quadratic equation, explain and determine how long did the object stay above 54 m?

2025 CGSS PRELIMS P2 Q1 

Quadratic Functions, Equations and Inequalities

<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 \(t=10\).

Exercise

<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 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.

<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 \(0 \le t \le 5\).

<p>Two particles, \(A\) and \(B\), leave a point \(O\) at the same time and 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 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 \(O\).

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 \(B\).

<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 \(T\).

<ol class="abc-list"></ol>
<p>A particle moves in a straight line so that, at time \(t\) seconds, its displacement, \(s\) metre, is given by \(s=10-5e^{-t}-\frac{1}{5}t\). Calculate the</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">initial velocity of the particle,</span></div></li><li><div class="list-item-content"><span class="list-item-text">value of time \(T\), when the particle is instantaneously at rest,</span></div></li><li><div class="list-item-content"><span class="list-item-text">acceleration of the particle at this instant,</span></div></li><li><div class="list-item-content"><span class="list-item-text">distance travelled from \(t=0\) to \(t=5\).</span></div></li></ol>

A particle moves in a straight line so that, at time \(t\) seconds, its displacement, \(s\) metre, is given by \(s=10-5e^{-t}-\frac{1}{5}t\). Calculate the

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

acceleration of the particle at this instant,

distance travelled from \(t=0\) to \(t=5\).

<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.

Given the Velocity Function

<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 starts from a fixed point \(A\) and travels in a straight line. The velocity, \(v\) m/s, of the particle, \(t\) s after leaving \(A\), is given by \(v = 2 + t - \sqrt{10t + 4}\).</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the acceleration of the particle when it is at instantaneous rest.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the total distance travelled by the particle between \(t = 0\) s and \(t = 7\) s.</span></div></li></ol>

A particle starts from a fixed point \(A\) and travels in a straight line. The velocity, \(v\) m/s, of the particle, \(t\) s after leaving \(A\), is given by \(v = 2 + t - \sqrt{10t + 4}\).

Find the acceleration of the particle when it is at instantaneous rest.

Find the total distance travelled by the particle between \(t = 0\) s and \(t = 7\) s.

<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>An object at \(A\), with an initial displacement of \(3\) m to the right from a fixed point \(O\), travels in a straight line so that its velocity, \(v\) ms\({}^{-1}\), is given by \(v = t^2 - 5t + 6\) where \(t\) is the time in seconds after leaving \(A\).</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the values of \(t\) when the object comes to an instantaneous rest.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the acceleration of the object at \(t = 5\) s.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Obtain an expression, in terms of \(t\), for the displacement of the object from \(O\) after \(t\) seconds.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the average speed of the object in the first \(5\) seconds.</span></div></li></ol>

An object at \(A\), with an initial displacement of \(3\) m to the right from a fixed point \(O\), travels in a straight line so that its velocity, \(v\) ms\({}^{-1}\), is given by \(v = t^2 - 5t + 6\) where \(t\) is the time in seconds after leaving \(A\).

Find the values of \(t\) when the object comes to an instantaneous rest.

Find the acceleration of the object at \(t = 5\) s.

Obtain an expression, in terms of \(t\), for the displacement of the object from \(O\) after \(t\) seconds.

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

Math Question Bank