3.02f Non-uniform acceleration: using differentiation and integration

7 A car driver makes a journey in a straight line from \(A\) to \(B\), starting from rest. The speed of the car increases to a maximum, then decreases until the car is at rest at \(B\). The distance travelled by the car \(t\) seconds after leaving \(A\) is \(0.0000117 \left( 400 t ^ { 3 } - 3 t ^ { 4 } \right)\) metres.

1. Find the distance \(A B\).
2. Find the maximum speed of the car.
3. Find the acceleration of the car
   1. as it starts from \(A\),
   2. as it arrives at \(B\).
   3. Sketch the velocity-time graph for the journey.

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.

6 A particle starts from rest at a point \(O\) and moves in a horizontal straight line. The velocity of the particle is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t \mathrm {~s}\) after leaving \(O\). For \(0 \leqslant t < 60\), the velocity is given by $$v = 0.05 t - 0.0005 t ^ { 2 }$$ The particle hits a wall at the instant when \(t = 60\), and reverses the direction of its motion. The particle subsequently comes to rest at the point \(A\) when \(t = 100\), and for \(60 < t \leqslant 100\) the velocity is given by $$v = 0.025 t - 2.5$$

1. Find the velocity of the particle immediately before it hits the wall, and its velocity immediately after its hits the wall.
2. Find the total distance travelled by the particle.
3. Find the maximum speed of the particle and sketch the particle's velocity-time graph for \(0 \leqslant t \leqslant 100\), showing the value of \(t\) for which the speed is greatest. \section*{[Question 7 is printed on the next page.]}

6 Two particles \(A\) and \(B\) start to move at the same instant from a point \(O\). The particles move in the same direction along the same straight line. The acceleration of \(A\) at time \(t \mathrm {~s}\) after starting to move is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), where \(a = 0.05 - 0.0002 t\).

1. Find A's velocity when \(t = 200\) and when \(t = 500\). \(B\) moves with constant acceleration for the first 200 s and has the same velocity as \(A\) when \(t = 200 . B\) moves with constant retardation from \(t = 200\) to \(t = 500\) and has the same velocity as \(A\) when \(t = 500\).
2. Find the distance between \(A\) and \(B\) when \(t = 500\).

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 \(O\).

6 A particle \(P\) moves in a straight line. It starts at a point \(O\) on the line and at time \(t\) s after leaving \(O\) it has a velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = 6 t ^ { 2 } - 30 t + 24\).

1. Find the set of values of \(t\) for which the acceleration of the particle is negative.
2. Find the distance between the two positions at which \(P\) is at instantaneous rest.
3. Find the two positive values of \(t\) at which \(P\) passes through \(O\).

2 A particle \(P\) moves in a straight line, starting from a point \(O\). At time \(t \mathrm {~s}\) after leaving \(O\), the velocity of \(P , v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), is given by \(v = 4 t ^ { 2 } - 8 t + 3\).

1. Find the two values of \(t\) at which \(P\) is at instantaneous rest.
2. Find the distance travelled by \(P\) between these two times.

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 velocity of \(P , v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), is given by \(v = q t + r t ^ { 2 }\), where \(q\) and \(r\) are constants. The particle has velocity \(4 \mathrm {~ms} ^ { - 1 }\) when \(t = 1\) and when \(t = 2\).

1. Show that, when \(t = 0.5\), the acceleration of \(P\) is \(4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
   ............................................................................................................................... .
2. Find the values of \(t\) when \(P\) is at instantaneous rest.
3. The particle is at \(O\) when \(t = 3\). Find the distance of \(P\) from \(O\) when \(t = 0\).

4 A particle \(P\) moves in a straight line starting from a point \(O\). At time \(t \mathrm {~s}\) after leaving \(O\), the velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), of \(P\) is given by \(v = ( 2 t - 5 ) ^ { 3 }\).

1. Find the values of \(t\) when the acceleration of \(P\) is \(54 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. Find an expression for the displacement of \(P\) from \(O\) at time \(t \mathrm {~s}\).

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.

5 A particle \(P\) moves in a straight line from a fixed point \(O\). The velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of \(P\) at time \(t \mathrm {~s}\) is given by $$v = t ^ { 2 } - 8 t + 12 \quad \text { for } 0 \leqslant t \leqslant 8$$

1. Find the minimum velocity of \(P\).
2. Find the total distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 8\). \includegraphics[max width=\textwidth, alt={}, center]{555678d3-f37d-4822-a005-de8c6094dc50-12_401_1102_260_520} Two particles \(A\) and \(B\), of masses 0.4 kg and 0.2 kg respectively, are connected by a light inextensible string. Particle \(A\) is held on a smooth plane inclined at an angle of \(\theta ^ { \circ }\) to the horizontal. The string passes over a small smooth pulley \(P\) fixed at the top of the plane, and \(B\) hangs freely 0.5 m above horizontal ground (see diagram). The particles are released from rest with both sections of the string taut.
3. Given that the system is in equilibrium, find \(\theta\).
4. It is given instead that \(\theta = 20\). In the subsequent motion particle \(A\) does not reach \(P\) and \(B\) remains at rest after reaching the ground.
   1. Find the tension in the string and the acceleration of the system.
   2. Find the speed of \(A\) at the instant \(B\) reaches the ground.
   3. Use an energy method to find the total distance \(A\) moves up the plane before coming to instantaneous rest.
      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 \(P\) 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.

7 \includegraphics[max width=\textwidth, alt={}, center]{5cba3e17-3979-4c22-a415-2cdd60f09289-4_547_1237_269_456} A tractor \(A\) starts from rest and travels along a straight road for 500 seconds. The velocity-time graph for the journey is shown above. This graph consists of three straight line segments. Find

1. the distance travelled by \(A\),
2. the initial acceleration of \(A\). Another tractor \(B\) starts from rest at the same instant as \(A\), and travels along the same road for 500 seconds. Its velocity \(t\) seconds after starting is \(\left( 0.06 t - 0.00012 t ^ { 2 } \right) \mathrm { m } \mathrm { s } ^ { - 1 }\). Find
3. how much greater \(B\) 's initial acceleration is than \(A\) 's,
4. how much further \(B\) has travelled than \(A\), at the instant when \(B\) 's velocity reaches its maximum.

7 A particle starts from rest at the point \(A\) and travels in a straight line until it reaches the point \(B\). The velocity of the particle \(t\) seconds after leaving \(A\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = 0.009 t ^ { 2 } - 0.0001 t ^ { 3 }\). Given that the velocity of the particle when it reaches \(B\) is zero, find

1. the time taken for the particle to travel from \(A\) to \(B\),
2. the distance \(A B\),
3. the maximum velocity of the particle.

6 A particle \(P\) starts from rest at \(O\) and travels in a straight line. Its velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t \mathrm {~s}\) is given by \(v = 8 t - 2 t ^ { 2 }\) for \(0 \leqslant t \leqslant 3\), and \(v = \frac { 54 } { t ^ { 2 } }\) for \(t > 3\). Find

1. the distance travelled by \(P\) in the first 3 seconds,
2. an expression in terms of \(t\) for the displacement of \(P\) from \(O\), valid for \(t > 3\),
3. the value of \(v\) when the displacement of \(P\) from \(O\) is 27 m .

4 The velocity of a particle \(t \mathrm {~s}\) after it starts from rest is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = 1.25 t - 0.05 t ^ { 2 }\). Find

1. the initial acceleration of the particle,
2. the displacement of the particle from its starting point at the instant when its acceleration is \(0.05 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

7 A motorcyclist starts from rest at \(A\) and travels in a straight line. For the first part of the motion, the motorcyclist's displacement \(x\) metres from \(A\) after \(t\) seconds is given by \(x = 0.6 t ^ { 2 } - 0.004 t ^ { 3 }\).

1. Show that the motorcyclist's acceleration is zero when \(t = 50\) and find the speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at this time. For \(t \geqslant 50\), the motorcyclist travels at constant speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the value of \(t\) for which the motorcyclist's average speed is \(27.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

4 A particle \(P\) starts from a fixed point \(O\) at time \(t = 0\), where \(t\) is in seconds, and moves with constant acceleration in a straight line. The initial velocity of \(P\) is \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and its velocity when \(t = 10\) is \(3.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the displacement of \(P\) from \(O\) when \(t = 10\). Another particle \(Q\) also starts from \(O\) when \(t = 0\) and moves along the same straight line as \(P\). The acceleration of \(Q\) at time \(t\) is \(0.03 t \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. Given that \(Q\) has the same velocity as \(P\) when \(t = 10\), show that it also has the same displacement from \(O\) as \(P\) when \(t = 10\).

4 A particle starts from rest at a point \(X\) and moves in a straight line until, 60 seconds later, it reaches a point \(Y\). At time \(t \mathrm {~s}\) after leaving \(X\), the acceleration of the particle is $$\begin{array} { r c c } 0.75 \mathrm {~m} \mathrm {~s} ^ { - 2 } & \text { for } & 0 < t < 4 \\ 0 \mathrm {~m} \mathrm {~s} ^ { - 2 } & \text { for } & 4 < t < 54 \\ - 0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 } & \text { for } & 54 < t < 60 \end{array}$$

1. Find the velocity of the particle when \(t = 4\) and when \(t = 60\), and sketch the velocity-time graph.
2. Find the distance \(X Y\).

6 A particle travels along a straight line. It starts from rest at a point \(A\) on the line and comes to rest again, 10 seconds later, at another point \(B\) on the line. The velocity \(t\) seconds after leaving \(A\) is $$\begin{array} { r l l } 0.72 t ^ { 2 } - 0.096 t ^ { 3 } & \text { for } & 0 \leqslant t \leqslant 5 \\ 2.4 t - 0.24 t ^ { 2 } & \text { for } & 5 \leqslant t \leqslant 10 \end{array}$$

1. Show that there is no instantaneous change in the acceleration of the particle when \(t = 5\).
2. Find the distance \(A B\).

7 A particle \(P\) starts from a point \(O\) and moves along a straight line. \(P\) 's velocity \(t\) s after leaving \(O\) is \(\nu \mathrm { m } \mathrm { s } ^ { - 1 }\), where $$v = 0.16 t ^ { \frac { 3 } { 2 } } - 0.016 t ^ { 2 } .$$ \(P\) comes to rest instantaneously at the point \(A\).

1. Verify that the value of \(t\) when \(P\) is at \(A\) is 100 .
2. Find the maximum speed of \(P\) in the interval \(0 < t < 100\).
3. Find the distance \(O A\).
4. Find the value of \(t\) when \(P\) passes through \(O\) on returning from \(A\).

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 \(t = 16\).

7 A tractor travels in a straight line from a point \(A\) to a point \(B\). The velocity of the tractor is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t \mathrm {~s}\) after leaving \(A\).

1. \includegraphics[max width=\textwidth, alt={}, center]{2bd9f770-65b1-48c2-bf58-24e732bb6988-4_668_1091_397_568} The diagram shows an approximate velocity-time graph for the motion of the tractor. The graph consists of two straight line segments. Use the graph to find an approximation for
   1. the distance \(A B\),
   2. the acceleration of the tractor for \(0 < t < 400\) and for \(400 < t < 800\).
   3. The actual velocity of the tractor is given by \(v = 0.04 t - 0.00005 t ^ { 2 }\) for \(0 \leqslant t \leqslant 800\).
      (a) Find the values of \(t\) for which the actual acceleration of the tractor is given correctly by the approximate velocity-time graph in part (i). For the interval \(0 \leqslant t \leqslant 400\), the approximate velocity of the tractor in part (i) is denoted by \(v _ { 1 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
      (b) Express \(v _ { 1 }\) in terms of \(t\) and hence show that \(v _ { 1 } - v = 0.00005 ( t - 200 ) ^ { 2 } - 1\).
   4. Deduce that \(- 1 \leqslant v _ { 1 } - v \leqslant 1\).

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 \(B\).