In this question, the units are metres and seconds. A particle \(P\) is travelling in a straight line through a fixed point \(O\).

At time \(t\) its acceleration, \(a\), is given by \(a=(2t-3)^2\), where \(t\geqslant0\). When \(t=3\), \(P\) has a velocity of \(6\).

(a)(i) Find an expression for the velocity, \(v\), of \(P\) at time \(t\).

(ii) Find the time when \(P\) is at rest.

When \(t=\frac52\), the displacement of \(P\) from \(O\) is \(4\).

(iii) Find the displacement of \(P\) from \(O\) when \(t=3\).

(b) Use calculus to find the approximate change in \(v\) when \(t\) increases from \(\frac52\) by the small amount \(0.02\).

A particle \(P\) moves in a straight line. \(t\) seconds after passing a fixed point, \(O\), the acceleration of \(P\), \(a\text{ m s}^{-2}\), is given by

\(a=t^2-2\) for \(0\leqslant t\leqslant4\).

\(a=19-5\mathrm{e}^{8-2t}\) for \(4\leqslant t\leqslant10\).

When \(t=3\), the velocity of \(P\) is \(-\frac13\text{ m s}^{-1}\) and its displacement from \(O\) is \(-\frac14\text{ m}\).

(a)(i) Find the velocity of \(P\) when \(t=4\).

(a)(ii) Find the displacement of \(P\) from \(O\) when \(t=4\).

(b) Find the displacement of \(P\) from \(O\) when \(t=10\).

A particle moves in a straight line. Its velocity, \(v \mathrm{~ms}^{-1}\), at time \(t\) seconds is given by \(v=\cos t-\sin t\) (a) Find the acceleration, \(a \mathrm{~ms}^{-2}\), when \(t=\frac{\pi}{3}\).

The displacement of the particle from a fixed point \(O\) at time \(t\) is \(s\) metres. The particle passes through \(O\) when \(t=0\). (b) Find the displacement at the time when the particle first changes direction after passing through \(O\). (c) Find an expression for \(a\) in terms of \(s\).

In this question, all distances are in metres and time, \(t\), is in seconds. A particle \(P\) is at a fixed point \(O\) at time \(t=0\). The velocity, \(v\), of \(P\) is given by \(v=3 \sin 2 t\) for \(t \geqslant 0\). (a) Find the exact value of \(t\) for which the velocity is zero for the first time after \(P\) leaves \(O\).

(b) Find an expression, in terms of \(t\), for the displacement of \(P\) from \(O\) at time \(t\).

(c) Find the distance travelled by \(P\) for \(0 \leqslant t \leqslant \pi\).

A particle moves in a straight line. At time \(t\) seconds after passing through a fixed point \(O\), its velocity, \(v\text{ m s}^{-1}\), is given by

\(v=10\sin2t-6\cos2t.\)

(a) Find an expression for the acceleration of the particle.

(b) Find the acceleration when \(t=\frac\pi4\).

(c) Find the first time at which the acceleration is zero.

(d) Find the displacement of the particle between \(t=\frac\pi4\) and \(t=\frac\pi2\).

The diagram shows the velocity-time graph for the motion of a particle over a period of 45 seconds. The velocity of the particle at \(t=30\) is \(V\text{ m s}^{-1}\). The distance travelled by the particle in the 45 seconds is 800 m.

(a) Find the value of \(V\).

(b) Find the acceleration of the particle when \(t=35\).

A particle travels in a straight line so that, \(t\) seconds after passing a fixed point, its velocity, \(v\text{ m s}^{-1}\), is given by

\(v=e^{t/4}\quad\text{for }0\leq t\leq4,\)

\(v=\frac{16e}{t^2}\quad\text{for }4\leq t\leq k.\)

The total distance travelled by the particle between \(t=0\) and \(t=k\) is \(13.4\) metres. Find the value of \(k\).

(a) A vehicle travels along a straight, horizontal road. At time \(t=0\) seconds, the vehicle, travelling at a velocity of \(w\,\mathrm{m\,s^{-1}}\), passes point \(O\). The vehicle travels at this constant velocity for \(12\) seconds. It then slows down, with constant deceleration, for \(10\) seconds until it reaches a velocity of \((w-14)\,\mathrm{m\,s^{-1}}\). It continues to travel at this velocity for \(28\) seconds until it reaches point \(A\), \(458\) m from \(O\).

Find the value of \(w\).

(b) A particle moves in a straight line. The velocity, \(v\,\mathrm{m\,s^{-1}}\), of the particle at time \(t\) seconds, where \(t\geq0\), is given by

\(v=(t-4)(t-5).\)

(i) Find the value of \(t\) for which the acceleration of the particle is \(0\,\mathrm{m\,s^{-2}}\).

(ii) Find the set of values of \(t\) for which the velocity of the particle is negative.

(iii) Find the distance travelled by the particle in the first \(5\) seconds of its motion.

A particle \(P\) moves in a straight line. Its acceleration, \(a\,\mathrm{m\,s^{-2}}\), is given by

\(a=6t\quad\text{for }0\leq t\leq3,\)

and

\(a=\frac{18\mathrm e^3}{\mathrm e^t}\quad\text{for }t\geq3.\)

When \(t=1\), the velocity of \(P\) is \(2\,\mathrm{m\,s^{-1}}\) and the displacement of \(P\) from \(O\) is \(-4\) metres.

(a)(i) Find the velocity of \(P\) when \(t=3\).

(a)(ii) Find the displacement of \(P\) from \(O\) when \(t=3\).

(b) Find an expression, in terms of \(t\), for the displacement of \(P\) from \(O\) when \(t\geq3\).

The velocity, \(v\text{ m s}^{-1}\), of a particle moving in a straight line, \(t\) seconds after passing through a fixed point \(O\), is given by

\(v=6\sin3t.\)

(a) Find the time at which the acceleration of the particle is first equal to \(-9\text{ m s}^{-2}\).

(b) Find the displacement of the particle from \(O\) when \(t=5.6\).

The acceleration, \(a\text{ m s}^{-2}\), of a particle at time \(t\) seconds is given by

\(a=-\frac{45}{(t+1)^2}.\)

When \(t=0\), the velocity of the particle is \(50\text{ m s}^{-1}\).

(a) Find an expression for the velocity of the particle in terms of \(t\).

(b) Find the distance travelled by the particle between \(t=1\) and \(t=10\).

A particle \(P\) travels in a straight line so that, \(t\) seconds after passing through a fixed point \(O\), its velocity, \(v\text{ ms}^{-1}\), is given by

\(v=\frac{t}{2e}\quad\text{for }0\leqslant t\leqslant2,\)

\(v=e^{-t/2}\quad\text{for }t\gt 2.\)

Given that, after leaving \(O\), particle \(P\) is never at rest, find the distance it travels between \(t=1\) and \(t=3\).

(a) The diagram shows the displacement-time graph for a runner, for \(0\leq t\leq40\).

(i) Find the distance the runner has travelled when \(t=40\).

(ii) On the axes, draw the corresponding velocity-time graph for the runner, for \(0\leq t\leq40\).

(b) A particle \(P\), moves in a straight line such that its displacement from a fixed point at time \(t\) is \(s\). The acceleration of \(P\) is given by

\((2t+4)^{-1/2},\quad t\gt0.\)

(i) Given that \(P\) has a velocity of \(9\) when \(t=6\), find the velocity of \(P\) at time \(t\).

(ii) Given that \(s=\frac13\) when \(t=6\), find the displacement of \(P\) at time \(t\).

A particle moves in a straight line so that, \(t\) seconds after passing through a fixed point \(O\), its velocity, \(v\text{ m s}^{-1}\), is given by

\(v=3t^2-30t+72.\)

(a) Find the distance between the particle's two positions of instantaneous rest.

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

A particle moves in a straight line such that its displacement, \(s\) metres, from a fixed point \(O\) at time \(t\) seconds, is given by

\(s=2+t-2\cos t,\quad t\geq0.\)

(a) Find the displacement of the particle from \(O\) at the time when it first comes to instantaneous rest.

(b) Find the time when the particle next comes to rest.

(c) Find the distance travelled by the particle for

\(0\leq t\leq\frac{3\pi}{2}.\)

(a) The diagram shows the velocity-time graph for a particle \(P\), travelling in a straight line with velocity \(v\text{ ms}^{-1}\) at a time \(t\) seconds. \(P\) accelerates at a constant rate for the first \(10\) seconds of its motion, and then travels at constant velocity, \(30\text{ ms}^{-1}\), for another \(15\) seconds. \(P\) then accelerates at a constant rate for a further \(10\) seconds and reaches a velocity of \(60\text{ ms}^{-1}\). \(P\) then decelerates at a constant rate and comes to rest when \(t=55\).

(i) Find the acceleration when \(t=12\).

(ii) Find the acceleration when \(t=50\).

(iii) Find the total distance travelled by the particle \(P\).

(b) A particle \(Q\) travels in a straight line such that its velocity, \(v\text{ ms}^{-1}\), at time \(t\) seconds after passing through a fixed point \(O\) is given by \(v=4\cos3t-4\).

(i) Find the speed of \(Q\) when \(t=\frac{5\pi}{9}\).

(ii) Find the smallest positive value of \(t\) for which the acceleration of \(Q\) is zero.

(iii) Find an expression for the displacement of \(Q\) from \(O\) at time \(t\).

A particle moves in a straight line. Its acceleration is constant and equal to \(-6\text{ m s}^{-2}\). Initially, the particle is at \(O\) and has velocity \(18\text{ m s}^{-1}\).

(a) Find the time at which the particle is instantaneously at rest.

(b) Find the distance travelled by the particle in the third second.

(a) A particle \(P\) moves in a straight line such that its displacement, \(x\) m, from a fixed point \(O\) at time \(t\) s is given by

\(x=10\sin2t-5.\)

(i) Find the speed of \(P\) when \(t=\pi\).

(ii) Find the value of \(t\) for which \(P\) is first at rest.

(iii) Find the acceleration of \(P\) when it is first at rest.

(b) The diagram shows the velocity-time graph for a particle \(Q\) travelling in a straight line with velocity \(v\text{ m s}^{-1}\) at time \(t\) s. The particle accelerates at \(3.5\text{ m s}^{-2}\) for the first \(10\) s of its motion and then travels at constant velocity, \(V\text{ m s}^{-1}\), for \(10\) s. The particle then decelerates at a constant rate and comes to rest. The distance travelled during the interval \(20\leq t\leq25\) is \(112.5\) m.

(i) Find the value of \(V\).

(ii) Find the velocity of \(Q\) when \(t=25\).

(iii) Find the value of \(t\) when \(Q\) comes to rest.

At time \(t\) s, a particle travelling in a straight line has acceleration

\((2t+1)^{-\frac12}\text{ m s}^{-2}.\)

When \(t=0\), the particle is \(4\) m from a fixed point \(O\) and is travelling with velocity \(8\text{ m s}^{-1}\) away from \(O\).

(a) Find the velocity of the particle at time \(t\) s.

(b) Find the displacement of the particle from \(O\) at time \(t\) s.

A particle travels in a straight line. As it passes through a fixed point \(O\), the particle is travelling at a velocity of \(3\text{ m s}^{-1}\). The particle continues at this velocity for 60 seconds then decelerates at a constant rate for 15 seconds to a velocity of \(1.6\text{ m s}^{-1}\). The particle then decelerates again at a constant rate for 5 seconds to reach point \(A\), where it stops.

(a) Sketch the velocity-time graph for this journey.

(b) Find the distance between \(O\) and \(A\).

(c) Find the deceleration in the last 5 seconds.

The diagram shows the velocity-time graph for the motion of a car during the first \(90\) seconds of a journey. The total distance travelled in this time is \(2775\) m.

(a)(i) Find the value of \(V\).

(a)(ii) Find the acceleration of the car when \(t=40\).

(b) The acceleration of a particle is given by

\(a=6\cos 2t.\)

When \(t=0\), the velocity is \(10\) and the displacement is \(0\). Find expressions for the velocity and displacement at time \(t\).

The velocity-time graph for a particle \(P\) travelling in a straight line is shown.

(a) Find

(i) the acceleration of \(P\) at \(t=5\),

(ii) the distance travelled by \(P\) in the first 10 seconds.

A second particle \(Q\) moves with velocity \(v=3\sin2t-1\), where \(t\) is measured in seconds and \(v\) in metres per second.

(b) Find

(i) the speed of \(Q\) when \(t=\dfrac{7\pi}{12}\),

(ii) the first positive value of \(t\) for which \(Q\) has zero acceleration.

The velocity-time graph represents the motion of a particle travelling in a straight line.

(i) Find the acceleration during the last 6 seconds of the motion.

(ii) The particle travels with constant velocity for 23 seconds. Find \(k\).

(iii) Using your answer to part (ii), find the total distance travelled by the particle.

The velocity \(v\text{ m s}^{-1}\) of a particle \(t\) seconds after passing through a fixed point \(O\) is given by \(v=\dfrac{4}{(t+1)^3}\).

(i) Explain why the particle never changes direction.

(ii) Find the acceleration of the particle when \(t=5\).

(iii) Find an expression for the displacement of the particle from \(O\) after \(t\) seconds.

(iv) Find the distance travelled by the particle in the fourth second.

(a) The velocity-time graph for a particle \(P\) is shown by the two straight lines in the diagram.

(i) Find the deceleration of \(P\) for \(5\leq t\leq10\).

(ii) Write down the value of \(t\) when the speed of \(P\) is zero.

(iii) Find the distance \(P\) has travelled for \(0\leq t\leq10\).

(b) A particle \(Q\) has a displacement of \(x\) m from a fixed point \(O\), \(t\) s after leaving \(O\). The velocity, \(v\text{ m s}^{-1}\), of \(Q\) at time \(t\) s is given by

\(v=6e^{2t}+1.\)

(i) Find an expression for \(x\) in terms of \(t\).

(ii) Find the value of \(t\) when the acceleration of \(Q\) is \(24\text{ m s}^{-2}\).

A particle \(P\) moves in a straight line. The displacement of \(P\) from a fixed point \(O\) is \(x\) metres at time \(t\) seconds. The particle passes through \(O\) when \(t=0\), and its velocity \(v\) metres per second is given by

\(v=12e^{2t}-48t,\qquad t\geq0.\)

(i) Find \(x\) in terms of \(t\).

(ii) Find the value of \(t\) for which the acceleration of \(P\) is zero.

(iii) Find the velocity of \(P\) when its acceleration is zero.

A particle moves in a straight line, such that its velocity, \(v\text{ ms}^{-1}\), \(t\) s after passing a fixed point \(O\), is given by

\(v=2+6t+3\sin2t.\)

(i) Find the acceleration of the particle at time \(t\).

(ii) Hence find the smallest value of \(t\) for which the acceleration of the particle is zero.

(iii) Find the displacement, \(x\) m from \(O\), of the particle at time \(t\).