Variable acceleration with initial conditions

5 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 \(t \mathrm {~s}\) after leaving \(A\) is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), where $$a = 6 t ^ { \frac { 1 } { 2 } } - 2 t$$

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

5 A particle starts from rest from a point \(O\) and moves in a straight line. The acceleration of the particle at time \(t \mathrm {~s}\) after leaving \(O\) is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), where \(a = k t ^ { \frac { 1 } { 2 } }\) for \(0 \leqslant t \leqslant 9\) and where \(k\) is a constant. The velocity of the particle at \(t = 9\) is \(1.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that \(k = 0.1\).
   For \(t > 9\), the velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of the particle is given by \(v = 0.2 ( t - 9 ) ^ { 2 } + 1.8\).
2. Show that the distance travelled in the first 9 seconds is one tenth of the distance travelled between \(t = 9\) and \(t = 18\).
3. Find the greatest acceleration of the particle during the first 10 seconds of its motion.

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.

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 \(O\).

5 A particle \(P\) moves on the \(x\)-axis from the origin \(O\) with an initial velocity of \(- 20 \mathrm {~ms} ^ { - 1 }\). The acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) at time \(t \mathrm {~s}\) after leaving \(O\) is given by \(a = 12 - 2 t\).

1. Sketch a velocity-time graph for \(0 \leqslant t \leqslant 12\), indicating the times when \(P\) is at rest.
2. Find the total distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 12\).
   \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{167f782c-3047-41f9-90a8-32ccdc19216d-10_410_723_260_717} \captionsetup{labelformat=empty} \caption{Fig. 6.1}
   \end{figure} Fig. 6.1 shows particles \(A\) and \(B\), of masses 4 kg and 3 kg respectively, attached to the ends of a light inextensible string that passes over a small smooth pulley. The pulley is fixed at the top of a plane which is inclined at an angle of \(30 ^ { \circ }\) to the horizontal. \(A\) hangs freely below the pulley and \(B\) is on the inclined plane. The string is taut and the section of the string between \(B\) and the pulley is parallel to a line of greatest slope of the plane.
3. It is given that the plane is rough and the particles are in limiting equilibrium. Find the coefficient of friction between \(B\) and the plane.
4. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{167f782c-3047-41f9-90a8-32ccdc19216d-11_412_899_276_589} \captionsetup{labelformat=empty} \caption{Fig. 6.2}
   \end{figure} It is given instead that the plane is smooth and the particles are released from rest when the difference in the vertical heights of the particles is 1 m (see Fig. 6.2). Use an energy method to find the speed of the particles at the instant when the particles are at the same horizontal level.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

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.

7 A particle travels in a straight line from \(A\) to \(B\) in 20 s . Its acceleration \(t\) seconds after leaving \(A\) is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), where \(a = \frac { 3 } { 160 } t ^ { 2 } - \frac { 1 } { 800 } t ^ { 3 }\). It is given that the particle comes to rest at \(B\).

1. Show that the initial speed of the particle is zero.
2. Find the maximum speed of the particle.
3. Find the distance \(A B\).

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.

6 A particle moves in a straight line. It starts from rest at a fixed point \(O\) on the line. Its acceleration at time \(t \mathrm {~s}\) after leaving \(O\) is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), where \(a = 0.4 t ^ { 3 } - 4.8 t ^ { \frac { 1 } { 2 } }\).

1. Show that, in the subsequent motion, the acceleration of the particle when it comes to instantaneous rest is \(16 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. Find the displacement of the particle from \(O\) at \(t = 5\).
   \includegraphics[max width=\textwidth, alt={}, center]{06df8c0d-dd38-4e3b-b1a4-72120a81050e-12_554_878_260_635} The diagram shows the vertical cross-section \(P Q R\) of a slide. The part \(P Q\) is a straight line of length 8 m inclined at angle \(\alpha\) to the horizontal, where \(\sin \alpha = 0.8\). The straight part \(P Q\) is tangential to the curved part \(Q R\), and \(R\) is \(h \mathrm {~m}\) above the level of \(P\). The straight part \(P Q\) of the slide is rough and the curved part \(Q R\) is smooth. A particle of mass 0.25 kg is projected with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from \(P\) towards \(Q\) and comes to rest at \(R\). The coefficient of friction between the particle and \(P Q\) is 0.5 .
3. Find the work done by the friction force during the motion of the particle from \(P\) to \(Q\).
4. Hence find the speed of the particle at \(Q\).
5. Find the value of \(h\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

5 A particle \(P\) moves in a straight line. It starts from rest at \(A\) and comes to rest instantaneously at \(B\). The velocity of \(P\) at time \(t\) seconds after leaving \(A\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = 6 t ^ { 2 } - k t ^ { 3 }\) and \(k\) is a constant.

1. Find an expression for the displacement of \(P\) from \(A\) in terms of \(t\) and \(k\).
2. Find an expression for \(t\) in terms of \(k\) when \(P\) is at \(B\). Given that the distance \(A B\) is 108 m , find
3. the value of \(k\),
4. the maximum value of \(v\) when the particle is moving from \(A\) towards \(B\).

7 A particle \(P\) starts to move from a point \(O\) and travels in a straight line. The velocity of \(P\) is \(k \left( 60 t ^ { 2 } - t ^ { 3 } \right) \mathrm { ms } ^ { - 1 }\) at time \(t \mathrm {~s}\) after leaving \(O\), where \(k\) is a constant. The maximum velocity of \(P\) is \(6.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that \(k = 0.0002\).
   \(P\) comes to instantaneous rest at a point \(A\) on the line. Find
2. the distance \(O A\),
3. the magnitude of the acceleration of \(P\) at \(A\),
4. the speed of \(P\) when it subsequently passes through \(O\).

7 A particle moves in a straight line starting from rest from a point \(O\). The acceleration of the particle at time \(t \mathrm {~s}\) after leaving \(O\) is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), where $$a = 5.4 - 1.62 t$$

1. Find the positive value of \(t\) at which the velocity of the particle is zero, giving your answer as an exact fraction.
2. Find the velocity of the particle at \(t = 10\) and sketch the velocity-time graph for the first ten seconds of the motion.
3. Find the total distance travelled during the first ten seconds of the motion.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

7 A particle moves in a straight line. The particle is initially at rest at a point \(O\) on the line. At time \(t \mathrm {~s}\) after leaving \(O\), the acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) of the particle is given by \(a = 25 - t ^ { 2 }\) for \(0 \leqslant t \leqslant 9\).

1. Find the maximum velocity of the particle in this time period.
2. Find the total distance travelled until the maximum velocity is reached.
   The acceleration of the particle for \(t > 9\) is given by \(a = - 3 t ^ { - \frac { 1 } { 2 } }\).
3. Find the velocity of the particle when \(t = 25\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

5 A particle starts from rest at a point \(A\) at time \(t = 0\), where \(t\) is in seconds. The particle moves in a straight line. For \(0 \leqslant t \leqslant 4\) the acceleration is \(1.8 t \mathrm {~m} \mathrm {~s} ^ { - 2 }\), and for \(4 \leqslant t \leqslant 7\) the particle has constant acceleration \(7.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Find an expression for the velocity of the particle in terms of \(t\), valid for \(0 \leqslant t \leqslant 4\).
2. Show that the displacement of the particle from \(A\) is 19.2 m when \(t = 4\).
3. Find the displacement of the particle from \(A\) when \(t = 7\).

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} (a) State which of these three graphs is most appropriate to represent the motion of the car.
   (b) For each of the two other graphs give a reason why it is not appropriate to represent the motion of the car.

6 A particle starts from rest at the point A and travels in a straight line. The displacement sm of the particle from A at time ts after leaving A is given by $$s = 0.001 t ^ { 4 } - 0.04 t ^ { 3 } + 0.6 t ^ { 2 } , \quad \text { for } 0 \leqslant t \leqslant 10 .$$

1. Show that the velocity of the particle is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when \(\mathrm { t } = 10\). The acceleration of the particle for \(t \geqslant 10\) is \(( 0.8 - 0.08 t ) \mathrm { m } \mathrm { s } ^ { - 2 }\).
2. Show that the velocity of the particle is zero when \(\mathrm { t } = 20\).
3. Find the displacement from A of the particle when \(\mathrm { t } = 20\).

5 A particle \(P\) moves in a straight line so that, at time \(t\) seconds after leaving a fixed point \(O\), its acceleration is \(- \frac { 1 } { 10 } t \mathrm {~m} \mathrm {~s} ^ { - 2 }\). At time \(t = 0\), the velocity of \(P\) is \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find, by integration, an expression in terms of \(t\) and \(V\) for the velocity of \(P\).
2. Find the value of \(V\), given that \(P\) is instantaneously at rest when \(t = 10\).
3. Find the displacement of \(P\) from \(O\) when \(t = 10\).
4. Find the speed with which the particle returns to \(O\).

3 Two girls, Marie and Nina, are members of an Olympic hockey team. They are doing fitness training. Marie runs along a straight line at a constant speed of \(6 \mathrm {~ms} ^ { - 1 }\). Nina is stationary at a point O on the line until Marie passes her. Nina immediately runs after Marie until she catches up with her. The time, \(t \mathrm {~s}\), is measured from the moment when Nina starts running. So when \(t = 0\), both girls are at O .
Nina's acceleration, \(a \mathrm {~ms} ^ { - 2 }\), is given by $$\begin{array} { l l } a = 4 - t & \text { for } 0 \leqslant t \leqslant 4 , \\ a = 0 & \text { for } t > 4 . \end{array}$$

1. Show that Nina's speed, \(v \mathrm {~ms} ^ { - 1 }\), is given by $$\begin{array} { l l } v = 4 t - \frac { 1 } { 2 } t ^ { 2 } & \text { for } 0 \leqslant t \leqslant 4 , \\ v = 8 & \text { for } t > 4 . \end{array}$$
2. Find an expression for the distance Nina has run at time \(t\), for \(0 \leqslant t \leqslant 4\). Find how far Nina has run when \(t = 4\) and when \(t = 5 \frac { 1 } { 3 }\).
3. Show that Nina catches up with Marie when \(t = 5 \frac { 1 } { 3 }\).

6 In this question the origin is a point on the ground. The directions of the unit vectors \(\left( \begin{array} { l } 1 \\ 0 \\ 0 \end{array} \right) , \left( \begin{array} { l } 0 \\ 1 \\ 0 \end{array} \right)\) and \(\left( \begin{array} { l } 0 \\ 0 \\ 1 \end{array} \right)\) are
\includegraphics[max width=\textwidth, alt={}, center]{63a2dc41-5e8b-4275-8653-ece5067c4306-5_398_689_434_689} Alesha does a sky-dive on a day when there is no wind. The dive starts when she steps out of a moving helicopter. The dive ends when she lands gently on the ground.

• During the dive Alesha can reduce the magnitude of her acceleration in the vertical direction by spreading her arms and increasing air resistance.
• During the dive she can use a power unit strapped to her back to give herself an acceleration in a horizontal direction.
• Alesha's mass, including her equipment, is 100 kg .
• Initially, her position vector is \(\left( \begin{array} { r } - 75 \\ 90 \\ 750 \end{array} \right) \mathrm { m }\) and her velocity is \(\left( \begin{array} { r } - 5 \\ 0 \\ - 10 \end{array} \right) \mathrm { ms } ^ { - 1 }\).
  1. Calculate Alesha's initial speed, and the initial angle between her motion and the downward vertical.

At a certain time during the dive, forces of \(\left( \begin{array} { r } 0 \\ 0 \\ - 980 \end{array} \right) \mathrm { N } , \left( \begin{array} { r } 0 \\ 0 \\ 880 \end{array} \right) \mathrm { N }\) and \(\left( \begin{array} { r } 50 \\ - 20 \\ 0 \end{array} \right) \mathrm { N }\) are acting on Alesha.

Suggest how these forces could arise.

Find Alesha's acceleration at this time, giving your answer in vector form, and show that, correct to 3 significant figures, its magnitude is \(1.14 \mathrm {~ms} ^ { - 2 }\). One suggested model for Alesha's motion is that the forces on her are constant throughout the dive from when she leaves the helicopter until she reaches the ground.

Find expressions for her velocity and position vector at time \(t\) seconds after the start of the dive according to this model. Verify that when \(t = 30\) she is at the origin.

Explain why consideration of Alesha's landing velocity shows this model to be unrealistic.

3. A particle \(P\) moves along the \(x\)-axis. At time \(t = 0 , P\) passes through the origin with speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive \(x\) direction. The acceleration of \(P\) at time \(t\) seconds, where \(t \geqslant 0\), is \(( 4 t - 8 ) \mathrm { m } \mathrm { s } ^ { - 2 }\) in the positive \(x\) direction.

1. 1. Show that \(P\) is instantaneously at rest when \(t = 1\)
   2. Find the other value of \(t\) for which \(P\) is instantaneously at rest.
2. Find the total distance travelled by \(P\) in the interval \(1 \leqslant t \leqslant 4\)