3.02f Non-uniform acceleration: using differentiation and integration

5 Particle \(P\) travels along a straight line from \(A\) to \(B\) with constant acceleration \(0.05 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Its speed at \(A\) is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and its speed at \(B\) is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the time taken for \(P\) to travel from \(A\) to \(B\), and find also the distance \(A B\). Particle \(Q\) also travels along the same straight line from \(A\) to \(B\), starting from rest at \(A\). At time \(t \mathrm {~s}\) after leaving \(A\), the speed of \(Q\) is \(k t ^ { 3 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(k\) is a constant. \(Q\) takes the same time to travel from \(A\) to \(B\) as \(P\) does.
2. Find the value of \(k\) and find \(Q\) 's speed at \(B\).

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 \(O\).

2 A particle moves in a straight line. Its velocity \(t\) seconds after leaving a fixed point \(O\) on the line is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = 0.2 t + 0.006 t ^ { 2 }\). For the instant when the acceleration of the particle is 2.5 times its initial acceleration,

1. show that \(t = 25\),
2. find the displacement of the particle from \(O\).

5 A particle \(P\) moves in a straight line. \(P\) starts from rest at \(O\) and travels to \(A\) where it comes to rest, taking 50 seconds. The speed of \(P\) at time \(t\) seconds after leaving \(O\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v\) is defined as follows. $$\begin{aligned} \text { For } 0 \leqslant t \leqslant 5 , & v = t - 0.1 t ^ { 2 } \\ \text { for } 5 \leqslant t \leqslant 45 , & v \text { is constant } \\ \text { for } 45 \leqslant t \leqslant 50 , & v = 9 t - 0.1 t ^ { 2 } - 200 \end{aligned}$$

1. Find the distance travelled by \(P\) in the first 5 seconds.
2. Find the total distance from \(O\) to \(A\), and deduce the average speed of \(P\) for the whole journey from \(O\) to \(A\).

7 A vehicle starts from rest at a point \(O\) and moves in a straight line. Its speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t\) seconds after leaving \(O\) is defined as follows. $$\begin{aligned} \text { For } 0 & \leqslant t \leqslant 60 , \quad v = k _ { 1 } t - 0.005 t ^ { 2 } \\ \text { for } t \geqslant 60 , \quad v & = \frac { k _ { 2 } } { \sqrt { } t } \end{aligned}$$ The distance travelled by the vehicle during the first 60 s is 540 m .

1. Find the value of the constant \(k _ { 1 }\) and show that \(k _ { 2 } = 12 \sqrt { } ( 60 )\).
2. Find an expression in terms of \(t\) for the total distance travelled when \(t \geqslant 60\).
3. Find the speed of the vehicle when it has travelled a total distance of 1260 m .

4 Particles \(P\) and \(Q\) move on a straight line \(A O B\). The particles leave \(O\) simultaneously, with \(P\) moving towards \(A\) and with \(Q\) moving towards \(B\). The initial speed of \(P\) is \(1.3 \mathrm {~ms} ^ { - 1 }\) and its acceleration in the direction \(O A\) is \(0.1 \mathrm {~m} \mathrm {~s} ^ { - 2 } . Q\) moves with acceleration in the direction \(O B\) of \(0.016 t \mathrm {~m} \mathrm {~s} ^ { - 2 }\), where \(t\) seconds is the time elapsed since the instant that \(P\) and \(Q\) started to move from \(O\). When \(t = 20\), particle \(P\) passes through \(A\) and particle \(Q\) passes through \(B\).

1. Given that the speed of \(Q\) at \(B\) is the same as the speed of \(P\) at \(A\), find the speed of \(Q\) at time \(t = 0\).
2. Find the distance \(A B\).

7 \includegraphics[max width=\textwidth, alt={}, center]{c7133fc4-9a14-43fd-b5ed-788da72291cd-4_512_1351_998_397} The diagram shows the velocity-time graph for the motion of a particle \(P\) which moves on a straight line \(B A C\). It starts at \(A\) and travels to \(B\) taking 5 s. It then reverses direction and travels from \(B\) to \(C\) taking 10 s . For the first 3 s of \(P\) 's motion its acceleration is constant. For the remaining 12 s the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t \mathrm {~s}\) after leaving \(A\), where $$v = - 0.2 t ^ { 2 } + 4 t - 15 \text { for } 3 \leqslant t \leqslant 15$$

1. Find the value of \(v\) when \(t = 3\) and the magnitude of the acceleration of \(P\) for the first 3 s of its motion.
2. Find the maximum velocity of \(P\) while it is moving from \(B\) to \(C\).
3. Find the average speed of \(P\),
   1. while moving from \(A\) to \(B\),
   2. for the whole journey.

6 A particle \(P\) moves in a straight line, starting from a point \(O\). The velocity of \(P\), measured in \(\mathrm { m } \mathrm { s } ^ { - 1 }\), at time \(t \mathrm {~s}\) after leaving \(O\) is given by $$v = 0.6 t - 0.03 t ^ { 2 }$$

1. Verify that, when \(t = 5\), the particle is 6.25 m from \(O\). Find the acceleration of the particle at this time.
2. Find the values of \(t\) at which the particle is travelling at half of its maximum velocity.

3 A particle \(P\) moves along a straight line for 100 s . It starts at a point \(O\) and at time \(t\) seconds after leaving \(O\) the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where $$v = 0.00004 t ^ { 3 } - 0.006 t ^ { 2 } + 0.288 t$$

1. Find the values of \(t\) at which the acceleration of \(P\) is zero.
2. Find the displacement of \(P\) from \(O\) when \(t = 100\).

6 A particle \(P\) starts from rest at a point \(O\) of a straight line and moves along the line. The displacement of the particle at time \(t \mathrm {~s}\) after leaving \(O\) is \(x \mathrm {~m}\), where $$x = 0.08 t ^ { 2 } - 0.0002 t ^ { 3 }$$

1. Find the value of \(t\) when \(P\) returns to \(O\) and find the speed of \(P\) as it passes through \(O\) on its return.
2. For the motion of \(P\) until the instant it returns to \(O\), find
   1. the total distance travelled,
   2. the average speed.

7 A racing car is moving in a straight line. The acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) at time \(t \mathrm {~s}\) after the car starts from rest is given by $$\begin{array} { l l } a = 15 t - 3 t ^ { 2 } & \text { for } 0 \leqslant t \leqslant 5 \\ a = - \frac { 625 } { t ^ { 2 } } & \text { for } 5 < t \leqslant k \end{array}$$ where \(k\) is a constant.

1. Find the maximum acceleration of the car in the first five seconds of its motion.
2. Find the distance of the car from its starting point when \(t = 5\).
3. The car comes to rest when \(t = k\). Find the value of \(k\).

2 A particle moves in a straight line. Its displacement \(t \mathrm {~s}\) after leaving a fixed point \(O\) on the line is \(s \mathrm {~m}\), where \(s = 2 t ^ { 2 } - \frac { 80 } { 3 } t ^ { \frac { 3 } { 2 } }\).

1. Find the time at which the acceleration of the particle is zero.
2. Find the displacement and velocity of the particle at this instant.

5 A particle \(P\) starts from a fixed point \(O\) and moves in a straight line. At time \(t\) s after leaving \(O\), the velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of \(P\) is given by \(v = 6 t - 0.3 t ^ { 2 }\). The particle comes to instantaneous rest at point \(X\).

1. Find the distance \(O X\). A second particle \(Q\) starts from rest from \(O\), at the same instant as \(P\), and also travels in a straight line. The acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) of \(Q\) is given by \(a = k - 12 t\), where \(k\) is a constant. The displacement of \(Q\) from \(O\) is 400 m when \(t = 10\).
2. Find the value of \(k\).

5 A particle starts from a point \(O\) and moves in a straight line. The velocity of the particle at time \(t \mathrm {~s}\) after leaving \(O\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where $$\begin{array} { l l } v = 1.5 + 0.4 t & \text { for } 0 \leqslant t \leqslant 5 , \\ v = \frac { 100 } { t ^ { 2 } } - 0.1 t & \text { for } t \geqslant 5 . \end{array}$$

1. Find the acceleration of the particle during the first 5 seconds of motion.
2. Find the value of \(t\) when the particle is instantaneously at rest.
3. Find the total distance travelled by the particle in the first 10 seconds of motion.

5 A particle starts from a fixed origin with velocity \(0.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and moves in a straight line. The acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) of the particle \(t \mathrm {~s}\) after it leaves the origin is given by \(a = k \left( 3 t ^ { 2 } - 12 t + 2 \right)\), where \(k\) is a constant. When \(t = 1\), the velocity of \(P\) is \(0.1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that the value of \(k\) is 0.1 .
2. Find an expression for the displacement of the particle from the origin in terms of \(t\).
3. Hence verify that the particle is again at the origin at \(t = 2\).

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.

6 Particle \(P\) travels in a straight line from \(A\) to \(B\). The velocity of \(P\) at time \(t \mathrm {~s}\) after leaving \(A\) is denoted by \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where $$v = 0.04 t ^ { 3 } + c t ^ { 2 } + k t$$ \(P\) takes 5 s to travel from \(A\) to \(B\) and it reaches \(B\) with speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The distance \(A B\) is 25 m .

1. Find the values of the constants \(c\) and \(k\).
2. Show that the acceleration of \(P\) is a minimum when \(t = 2.5\).

5 The acceleration of a particle moving in a straight line is \(( x - 2.4 ) \mathrm { m } \mathrm { s } ^ { - 2 }\) when its displacement from a fixed point \(O\) of the line is \(x \mathrm {~m}\). The velocity of the particle is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and it is given that \(v = 2.5\) when \(x = 0\). Find

1. an expression for \(v\) in terms of \(x\),
2. the minimum value of \(v\).

4 An object of mass 0.4 kg is projected vertically upwards from the ground, with an initial speed of \(16 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A resisting force of magnitude \(0.1 v\) newtons acts on the object during its ascent, where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the object at time \(t \mathrm {~s}\) after it starts to move.

1. Show that \(\frac { \mathrm { d } v } { \mathrm {~d} t } = - 0.25 ( v + 40 )\).
2. Find the value of \(t\) at the instant that the object reaches its maximum height.

2 A particle starts from rest at \(O\) and travels in a straight line. Its acceleration is \(( 3 - 2 x ) \mathrm { ms } ^ { - 2 }\), where \(x \mathrm {~m}\) is the displacement of the particle from \(O\).

1. Find the value of \(x\) for which the velocity of the particle reaches its maximum value.
2. Find this maximum velocity.

3 A particle \(P\) starts from a fixed point \(O\) and moves in a straight line. When the displacement of \(P\) from \(O\) is \(x \mathrm {~m}\), its velocity is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and its acceleration is \(\frac { 1 } { x + 2 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Given that \(v = 2\) when \(x = 0\), use integration to show that \(v ^ { 2 } = 2 \ln \left( \frac { 1 } { 2 } x + 1 \right) + 4\).
2. Find the value of \(v\) when the acceleration of \(P\) is \(\frac { 1 } { 4 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

7 A particle \(P\) of mass 0.25 kg moves in a straight line on a smooth horizontal surface. \(P\) starts at the point \(O\) with speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and moves towards a fixed point \(A\) on the line. At time \(t \mathrm {~s}\) the displacement of \(P\) from \(O\) is \(x \mathrm {~m}\) and the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A resistive force of magnitude (5-x) N acts on \(P\) in the direction towards \(O\).

1. Form a differential equation in \(v\) and \(x\). By solving this differential equation, show that \(v = 10 - 2 x\).
2. Find \(x\) in terms of \(t\), and hence show that the particle is always less than 5 m from \(O\).

4 A particle of mass 0.2 kg is projected vertically downwards with initial speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A resisting force of magnitude \(0.09 v \mathrm {~N}\) acts vertically upwards on the particle during its descent, where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the downwards velocity of the particle at time \(t \mathrm {~s}\) after being set in motion.

1. Show that the acceleration of the particle is \(( 10 - 0.45 v ) \mathrm { m } \mathrm { s } ^ { - 2 }\).
2. Find \(v\) when \(t = 1.5\).

7 A particle \(P\) of mass 0.5 kg moves in a straight line on a smooth horizontal surface. The velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when the displacement of \(P\) from \(O\) is \(x \mathrm {~m}\). A single horizontal force of magnitude \(0.16 \mathrm { e } ^ { x } \mathrm {~N}\) acts on \(P\) in the direction \(O P\). The velocity of \(P\) when it is at \(O\) is \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that \(v = 0.8 \mathrm { e } ^ { \frac { 1 } { 2 } x }\).
2. Find the time taken by \(P\) to travel 1.4 m from \(O\).