Variable acceleration with initial conditions

6 A cyclist starts from rest at a fixed point \(O\) and moves in a straight line, before coming to rest \(k\) seconds later. The acceleration of the cyclist at time \(t \mathrm {~s}\) after leaving \(O\) is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), where \(a = 2 t ^ { - \frac { 1 } { 2 } } - \frac { 3 } { 5 } t ^ { \frac { 1 } { 2 } }\) for \(0 < t \leqslant k\).

1. Find the value of \(k\).
2. Find the maximum speed of the cyclist.
3. Find an expression for the displacement from \(O\) in terms of \(t\). Hence find the total distance travelled by the cyclist from the time at which she reaches her maximum speed until she comes to rest.

7 A particle \(P\) moves in a straight line, starting from a point \(O\) with velocity \(1.72 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) of the particle, \(t \mathrm {~s}\) after leaving \(O\), is given by \(a = 0.1 t ^ { \frac { 3 } { 2 } }\).

1. Find the value of \(t\) when the velocity of \(P\) is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the displacement of \(P\) from \(O\) when \(t = 2\), giving your answer correct to 2 decimal places. [3]

5 A particle \(P\) moves in a straight line, starting from rest at a point \(O\) on the line. At time \(t \mathrm {~s}\) after leaving \(O\) the acceleration of \(P\) is \(k \left( 16 - t ^ { 2 } \right) \mathrm { m } \mathrm { s } ^ { - 2 }\), where \(k\) is a positive constant, and the displacement from \(O\) is \(s \mathrm {~m}\). The velocity of \(P\) is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when \(t = 4\).

1. Show that \(s = \frac { 1 } { 64 } t ^ { 2 } \left( 96 - t ^ { 2 } \right)\).
2. Find the speed of \(P\) at the instant that it returns to \(O\).
3. Find the maximum displacement of the particle from \(O\).

6 A particle moves in a straight line. At time \(t \mathrm {~s}\), the acceleration, \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), of the particle is given by \(a = 36 - 6 t\). The velocity of the particle is \(27 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when \(t = 2\).

1. Find the values of \(t\) when the particle is at instantaneous rest.
2. Find the total distance the particle travels during the first 12 seconds.

3 A particle \(P\) moves in a straight line, starting from the point \(O\) with velocity \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The acceleration of \(P\) at time \(t \mathrm {~s}\) after leaving \(O\) is \(2 t ^ { \frac { 2 } { 3 } } \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Show that \(t ^ { \frac { 5 } { 3 } } = \frac { 5 } { 6 }\) when the velocity of \(P\) is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the distance of \(P\) from \(O\) when the velocity of \(P\) is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

4 An aeroplane moves along a straight horizontal runway before taking off. It starts from rest at \(O\) and has speed \(90 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the instant it takes off. While the aeroplane is on the runway at time \(t\) seconds after leaving \(O\), its acceleration is \(( 1.5 + 0.012 t ) \mathrm { m } \mathrm { s } ^ { - 2 }\). Find

1. the value of \(t\) at the instant the aeroplane takes off,
2. the distance travelled by the aeroplane on the runway.

4 A particle \(P\) moves in a straight line. At time \(t\) seconds after starting from rest at the point \(O\) on the line, the acceleration of \(P\) is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), where \(a = 0.075 t ^ { 2 } - 1.5 t + 5\).

1. Find an expression for the displacement of \(P\) from \(O\) in terms of \(t\).
2. Hence find the time taken for \(P\) to return to the point \(O\).

7 A particle \(P\) moves in a straight line. At time \(t \mathrm {~s}\), the displacement of \(P\) from \(O\) is \(s \mathrm {~m}\) and the acceleration of \(P\) is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), where \(a = 6 t - 2\). When \(t = 1 , s = 7\) and when \(t = 3 , s = 29\).

1. Find the set of values of \(t\) for which the particle is decelerating.
2. Find \(s\) in terms of \(t\).
3. Find the time when the velocity of the particle is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

6 A particle \(P\) moves in a straight line passing through a point \(O\). At time \(t \mathrm {~s}\), the acceleration, \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), of \(P\) is given by \(a = 6 - 0.24 t\). The particle comes to instantaneous rest at time \(t = 20\).

1. Find the value of \(t\) at which the particle is again at instantaneous rest.
2. Find the distance the particle travels between the times of instantaneous rest.

6 A particle \(P\) moves in a straight line. The acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) of \(P\) at time \(t \mathrm {~s}\) is given by \(a = 6 t - 12\). The displacement of \(P\) from a fixed point \(O\) on the line is \(s \mathrm {~m}\). It is given that \(s = 5\) when \(t = 1\) and \(s = 1\) when \(t = 3\).

1. Show that \(s = t ^ { 3 } - 6 t ^ { 2 } + p t + q\), where \(p\) and \(q\) are constants to be found.
2. Find the values of \(t\) when \(P\) is at instantaneous rest.
3. Find the total distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 4\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

3 A particle \(P\) moves in a straight line. It starts from a point \(O\) on the line with velocity \(1.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The acceleration of \(P\) at time \(t \mathrm {~s}\) after leaving \(O\) is \(0.8 t ^ { - 0.75 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find the displacement of \(P\) from \(O\) when \(t = 16\).

8. \begin{figure}[h] \end{figure} The acceleration-time graph shown in Figure 5 represents part of a journey made by a car along a straight horizontal road. The car accelerated from rest at time \(t = 0\)

1. Find the distance travelled by the car during the first 4 s of its journey.
2. Find the total distance travelled by the car during the first 26s of its journey.
   

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5 A car is travelling at \(13 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along a straight road when it passes a point \(A\) at time \(t = 0\), where \(t\) is in seconds. For \(0 \leqslant t \leqslant 6\), the car accelerates at \(0.8 t \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Calculate the speed of the car when \(t = 6\).
2. Calculate the displacement of the car from \(A\) when \(t = 6\).
3. Three \(( t , x )\) graphs are shown below, for \(0 \leqslant t \leqslant 6\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{470e70de-66ba-4dcc-a205-0c92f29471b1-3_382_458_1366_340} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{470e70de-66ba-4dcc-a205-0c92f29471b1-3_382_460_1366_881} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{470e70de-66ba-4dcc-a205-0c92f29471b1-3_384_461_1366_1420} \captionsetup{labelformat=empty} \caption{Fig. 3}
   \end{figure}
   1. State which of these three graphs is most appropriate to represent the motion of the car.
   2. For each of the two other graphs give a reason why it is not appropriate to represent the motion of the car.

3. A particle moves along the \(x\)-axis. At time \(t = 0\) the particle passes through the origin with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive \(x\)-direction. The acceleration of the particle at time \(t\) seconds, \(t \geqslant 0\), is \(\left( 4 t ^ { 3 } - 12 t \right) \mathrm { m } \mathrm { s } ^ { - 2 }\) in the positive \(x\)-direction. Find

1. the velocity of the particle at time \(t\) seconds,
2. the displacement of the particle from the origin at time \(t\) seconds,
3. the values of \(t\) at which the particle is instantaneously at rest.

2 A particle is moving along a straight line and its position is relative to an origin on the line. At time \(t \mathrm {~s}\), the particle's acceleration, \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), is given by $$a = 6 t - 12$$ At \(t = 0\) the velocity of the particle is \(+ 9 \mathrm {~ms} ^ { - 1 }\) and its position is - 2 m .

1. Find an expression for the velocity of the particle at time \(t \mathrm {~s}\) and verify that it is stationary when \(t = 3\).
2. Find the position of the particle when \(t = 2\).

3 A particle is moving along a straight line and its position is relative to an origin on the line. At time \(t \mathrm {~s}\), the particle's acceleration, \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), is given by $$a = 6 t - 12 .$$ At \(t = 0\) the velocity of the particle is \(+ 9 \mathrm {~ms} ^ { - 1 }\) and its position is - 2 m .

1. Find an expression for the velocity of the particle at time \(t \mathrm {~s}\) and verify that it is stationary when \(t = 3\).
2. Find the position of the particle when \(t = 2\).

5 A particle is moving along a straight line and its position is relative to an origin on the line. At time \(t \mathrm {~s}\), the particle's acceleration, \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), is given by $$a = 6 t - 12 .$$ At \(t = 0\) the velocity of the particle is \(+ 9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and its position is - 2 m .

1. Find an expression for the velocity of the particle at time \(t \mathrm {~s}\) and verify that it is stationary when \(t = 3\).
2. Find the position of the particle when \(t = 2\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{b9e41fac-9f4b-4165-af03-67ebdcb326de-3_349_987_375_623} \captionsetup{labelformat=empty} \caption{Fig. 4}
   \end{figure} Particles P and Q move in the same straight line. Particle P starts from rest and has a constant acceleration towards \(Q\) of \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Particle \(Q\) starts 125 m from \(P\) at the same time and has a constant speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) away from \(P\). The initial values are shown in Fig. 4.
3. Write down expressions for the distances travelled by P and by Q at time \(t\) seconds after the start of the motion.
4. How much time does it take for P to catch up with Q and how far does P travel in this time?

4. A particle \(P\) moves in a straight horizontal line such that its acceleration at time \(t\) seconds is proportional to \(\left( 3 t ^ { 2 } - 5 \right)\). Given that at time \(t = 0 , P\) is at rest at the origin \(O\) and that at time \(t = 3\), its velocity is \(3 \mathrm {~ms} ^ { - 1 }\),

1. find, in \(\mathrm { m } \mathrm { s } ^ { - 2 }\), the acceleration of \(P\) in terms of \(t\),
2. show that the displacement of the particle, \(s\) metres, from \(O\) at time \(t\) is given by $$s = \frac { 1 } { 16 } t ^ { 2 } \left( t ^ { 2 } - 10 \right)$$ (4 marks)

3. At time \(t\) seconds the acceleration, \(a \mathrm {~ms} ^ { - 2 }\), of a particle is given by $$a = \frac { 4 } { ( 1 + t ) ^ { 3 } }$$ When \(t = 0\), the particle has velocity \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and displacement 3 m from a fixed origin \(O\).

1. Find an expression for the velocity of the particle in terms of \(t\).
2. Show that when \(t = 3\) the particle is 10.5 m from \(O\).

2 A particle P of mass \(m\) travels in a straight line on a smooth horizontal surface.
At time \(t , \mathrm { P }\) is a distance \(x\) from a fixed point O and is moving with speed \(v\) away from O . A horizontal force of magnitude \(3 m t\) acts on P , in a direction away from O .

1. Show that \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = 3 t\).
2. Verify that the general solution of this differential equation is \(x = \frac { 1 } { 2 } t ^ { 3 } + A t + k\), where \(A\) and \(k\) are constants.
3. Given that \(x = 6\) and \(v = 12\) when \(t = 1\), find the values of \(A\) and \(k\).

A particle moving in a straight line starts from rest at a point \(A\) and comes instantaneously to rest at a point \(B\). The acceleration of the particle at time \(t\) s after leaving \(A\) is \(a \text{ m s}^{-2}\), where $$a = 6t^{\frac{1}{2}} - 2t.$$

1. Find the value of \(t\) at point \(B\). [3]
2. Find the distance travelled from \(A\) to the point at which the acceleration of the particle is again zero. [5]

A particle moves in a straight line starting from rest from a point \(O\). The acceleration of the particle at time \(t\) s after leaving \(O\) is \(a\,\text{m}\,\text{s}^{-2}\), where \(a = 4t^2\).

1. Find the speed of the particle when \(t = 9\). [2]
2. Find the time after leaving \(O\) at which the speed (in metres per second) and the distance travelled (in metres) are numerically equal. [3]

A particle \(P\) moves in a straight line. It starts from rest at a point \(O\) on the line and at time \(t\) s after leaving \(O\) it has acceleration \(a \text{ m s}^{-2}\), where \(a = 6t - 18\). Find the distance \(P\) moves before it comes to instantaneous rest. [6]

A particle moves in a straight line. It starts from rest at a fixed point \(O\) on the line. Its acceleration at time \(t\) s after leaving \(O\) is \(a\) m s\(^{-2}\), where \(a = 0.4t^3 - 4.8t^2\).

1. Show that, in the subsequent motion, the acceleration of the particle when it comes to instantaneous rest is \(16\) m s\(^{-2}\). [6]
2. Find the displacement of the particle from \(O\) at \(t = 5\). [3]