3.02f Non-uniform acceleration: using differentiation and integration

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 \(0 \leq t \leq 4\) the acceleration is \(1.8t \text{ m s}^{-2}\), and for \(4 \leq t \leq 7\) the particle has constant acceleration \(7.2 \text{ m s}^{-2}\).

1. Find an expression for the velocity of the particle in terms of \(t\), valid for \(0 \leq t \leq 4\). [3]
2. Show that the displacement of the particle from \(A\) is 19.2 m when \(t = 4\). [4]
3. Find the displacement of the particle from \(A\) when \(t = 7\). [5]

\includegraphics{figure_7} A sprinter \(S\) starts from rest at time \(t = 0\), where \(t\) is in seconds, and runs in a straight line. For \(0 \leq t \leq 3\), \(S\) has velocity \((6t - t^2)\) m s\(^{-1}\). For \(3 < t \leq 22\), \(S\) runs at a constant speed of \(9\) m s\(^{-1}\). For \(t > 22\), \(S\) decelerates at \(0.6\) m s\(^{-2}\) (see diagram).

1. Express the acceleration of \(S\) during the first \(3\) seconds in terms of \(t\). [2]
2. Show that \(S\) runs \(18\) m in the first \(3\) seconds of motion. [5]
3. Calculate the time \(S\) takes to run \(100\) m. [3]
4. Calculate the time \(S\) takes to run \(200\) m. [7]

\includegraphics{figure_7} The diagram shows the \((t, v)\) graphs for two particles \(A\) and \(B\) which move on the same straight line. The units of \(v\) and \(t\) are \(\text{m s}^{-1}\) and \(\text{s}\) respectively. Both particles are at the point \(S\) on the line when \(t = 0\). The particle \(A\) is initially at rest, and moves with acceleration \(0.18t\text{ m s}^{-2}\) until the two particles collide when \(t = 16\). The initial velocity of \(B\) is \(U\text{ m s}^{-1}\) and \(B\) has variable acceleration for the first five seconds of its motion. For the next ten seconds of its motion \(B\) has a constant velocity of \(9\text{ m s}^{-1}\); finally \(B\) moves with constant deceleration for one second before it collides with \(A\).

1. Calculate the value of \(t\) at which the two particles have the same velocity. [4]

For \(0 \leq t \leq 5\) the distance of \(B\) from \(S\) is \((Ut + 0.08t^2)\text{ m}\).

1. Calculate \(U\) and verify that when \(t = 5\), \(B\) is \(25\text{ m}\) from \(S\). [4]
2. Calculate the velocity of \(B\) when \(t = 16\). [5]

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

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

\includegraphics{figure_5} A toy car is moving along the straight line \(Ox\), where O is the origin. The time \(t\) is in seconds. At time \(t = 0\) the car is at A, 3 m from O as shown in Fig. 5. The velocity of the car, \(v\) m s\(^{-1}\), is given by $$v = 2 + 12t - 3t^2.$$ Calculate the distance of the car from O when its acceleration is zero. [8]

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 \(6\) 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, \(t\) s, is measured from the moment when Nina starts running. So when \(t = 0\), both girls are at O. Nina's acceleration, \(a\) ms\(^{-2}\), is given by \begin{align} a &= 4 - t \quad \text{for } 0 < t < 4,
a &= 0 \quad \text{for } t > 4. \end{align}

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

A toy boat moves in a horizontal plane with position vector \(\mathbf{r} = x\mathbf{i} + y\mathbf{j}\), where \(\mathbf{i}\) and \(\mathbf{j}\) are the standard unit vectors east and north respectively. The origin of the position vectors is at O. The displacements \(x\) and \(y\) are in metres. First consider only the motion of the boat parallel to the \(x\)-axis. For this motion $$x = 8t - 2t^2.$$ The velocity of the boat in the \(x\)-direction is \(v_x\) ms\(^{-1}\).

1. Find an expression in terms of \(t\) for \(v_x\) and determine when the boat instantaneously has zero speed in the \(x\)-direction. [3]

Now consider only the motion of the boat parallel to the \(y\)-axis. For this motion $$v_y = (t - 2)(3t - 2),$$ where \(v_y\) ms\(^{-1}\) is the velocity of the boat in the \(y\)-direction at time \(t\) seconds.

1. Given that \(y = 3\) when \(t = 1\), use integration to show that \(y = t^3 - 4t^2 + 4t + 2\). [4]

The position vector of the boat is given in terms of \(t\) by \(\mathbf{r} = (8t - 2t^2)\mathbf{i} + (t^3 - 4t^2 + 4t + 2)\mathbf{j}\).

1. Find the time(s) when the boat is due north of O and also the distance of the boat from O at any such times. [4]
2. Find the time(s) when the boat is instantaneously at rest. Find the distance of the boat from O at any such times. [5]
3. Plot a graph of the path of the boat for \(0 \leq t \leq 2\). [3]

A box of emergency supplies is dropped to victims of a natural disaster from a stationary helicopter at a height of 1000 metres. The initial velocity of the box is zero. At time \(t\) s after being dropped, the acceleration, \(a\text{ m s}^{-2}\), of the box in the vertically downwards direction is modelled by $$a = 10 - t \text{ for } 0 \leqslant t \leqslant 10,$$ $$a = 0 \text{ for } t > 10.$$

1. Find an expression for the velocity, \(v\text{ m s}^{-1}\), of the box in the vertically downwards direction in terms of \(t\) for \(0 \leqslant t \leqslant 10\). Show that for \(t > 10\), \(v = 50\). [4]
2. Draw a sketch graph of \(v\) against \(t\) for \(0 \leqslant t \leqslant 20\). [3]
3. Show that the height, \(h\) m, of the box above the ground at time \(t\) s is given, for \(0 \leqslant t \leqslant 10\), by $$h = 1000 - 5t^2 + \frac{1}{6}t^3.$$ Find the height of the box when \(t = 10\). [4]
4. Find the value of \(t\) when the box hits the ground. [2]
5. Some of the supplies in the box are damaged when the box hits the ground. So measures are considered to reduce the speed with which the box hits the ground the next time one is dropped. Two different proposals are made. Carry out suitable calculations and then comment on each of them.
   1. The box should be dropped from a height of 500 m instead of 1000 m. [2]
   2. The box should be fitted with a parachute so that its acceleration is given by $$a = 10 - 2t \text{ for } 0 \leqslant t \leqslant 5,$$ $$a = 0 \text{ for } t > 5.$$ [3]

A particle \(P\), of mass 0.4 kg, moves in a straight line such that, at time \(t\) seconds after passing through a fixed point \(O\), its distance from \(O\) is \(x\) metres, where \(x = 3t^2 + 8t\).

1. Show that \(P\) never returns to \(O\). [2 marks]
2. Find the value of \(t\) when \(P\) has velocity 20 ms\(^{-1}\). [3 marks]
3. Show that the force acting on \(P\) is constant, and find its magnitude. [3 marks]

A particle \(P\), initially at rest at the point \(O\), moves in a straight line such that at time \(t\) seconds after leaving \(O\) its acceleration is \((12t - 15)\) ms\(^{-2}\). Find

1. the velocity of \(P\) at time \(t\) seconds after it leaves \(O\), [3 marks]
2. the value of \(t\) when the speed of \(P\) is 36 ms\(^{-1}\). [3 marks]

A rocket is fired from a fixed point \(O\). During the first phase of its motion its velocity, \(v\) ms\(^{-1}\), is given at time \(t\) seconds after firing by the formula $$v = pt^2 + qt.$$ \(5\) seconds after firing, the rocket is travelling at \(500\) ms\(^{-1}\). \(30\) seconds after firing, the rocket is travelling at \(12\,000\) ms\(^{-1}\).

1. Find the constants \(p\) and \(q\). [4 marks]
2. Sketch a velocity-time graph for the rocket for \(0 \leq t \leq 30\). [2 marks]
3. Find the initial acceleration of the rocket. [2 marks]
4. Find the distance of the rocket from \(O\) \(30\) seconds after firing. [4 marks]

From time \(t = 30\) onwards, the rocket maintains a constant speed of \(12\,000\) ms\(^{-1}\).

1. Find the average speed of the rocket during its first \(50\) seconds of motion. [3 marks]

The acceleration of a particle \(P\) is \((8t - 18)\) ms\(^{-2}\), where \(t\) seconds is the time that has elapsed since \(P\) passed through a fixed point \(O\) on the straight line on which it is moving. At time \(t = 3\), \(P\) has speed \(2\) ms\(^{-1}\). Find

1. the velocity of \(P\) at time \(t\), [4 marks]
2. the values of \(t\) when \(P\) is instantaneously at rest. [3 marks]

A particle \(P\) moves in a straight line so that its velocity \(v\) ms\(^{-1}\) at time \(t\) seconds is given, for \(t > 1\), by the formula \(v = 2t + \frac{8}{t^2}\). Find the time when the acceleration of \(P\) is zero. [5 marks]

The velocity, \(v\) ms\(^{-1}\), of a particle at time \(t\) s is given by \(v = 4t^2 - 9\).

1. Find the acceleration of the particle when it is instantaneously at rest. [3 marks]
2. Find the distance travelled by the particle from time \(t = 0\) until it comes to rest. [4 marks]

A particle of mass \(0.4\) kg, moving on a smooth horizontal surface, passes through a point \(O\) with velocity \(10\text{ ms}^{-1}\). At time \(t\) s after the particle passes through \(O\), the particle has a displacement \(x\) m from \(O\), has a velocity \(v\text{ ms}^{-1}\) away from \(O\), and is acted on by a force of magnitude \(\frac{1}{5}v\) N acting towards \(O\). Find

1. the time taken for the velocity of the particle to reduce from \(10\text{ ms}^{-1}\) to \(5\text{ ms}^{-1}\), [5]
2. the average velocity of the particle over this time. [6]

A particle \(Q\) of mass \(0.2\) kg is projected horizontally with velocity \(4\) m s\(^{-1}\) from a fixed point \(A\) on a smooth horizontal surface. At time \(t\) s after projection \(Q\) is \(x\) m from \(A\) and is moving away from \(A\) with velocity \(v\) m s\(^{-1}\). There is a force of \(3\cos 2t\) N acting on \(Q\) in the positive \(x\)-direction.

1. Find an expression for the velocity of \(Q\) at time \(t\). State the maximum and minimum values of the velocity of \(Q\) as \(t\) varies. [4]
2. Find the average velocity of \(Q\) between times \(t = \pi\) and \(t = \frac{3}{2}\pi\). [4]

A car is travelling on a straight horizontal road. The velocity of the car, \(v\) ms\(^{-1}\), at time \(t\) seconds as it travels past three points, \(P\), \(Q\) and \(R\), is modelled by the equation \(v = at^2 + bt + c\), where \(a\), \(b\) and \(c\) are constants. The car passes \(P\) at time \(t = 0\) with velocity \(8\) ms\(^{-1}\).

1. State the value of \(c\). [1] The car passes \(Q\) at time \(t = 5\) and at that instant its deceleration is \(0.12\) ms\(^{-2}\). The car passes \(R\) at time \(t = 18\) with velocity \(2.96\) ms\(^{-1}\).
2. Determine the values of \(a\) and \(b\). [4]
3. Find, to the nearest metre, the distance between points \(P\) and \(R\). [2]

In this question the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are in the directions east and north respectively. A particle \(P\) is moving on a smooth horizontal surface under the action of a single force \(\mathbf{F}\) N. At time \(t\) seconds, where \(t \geq 0\), the velocity \(\mathbf{v} \mathrm{m s}^{-1}\) of \(P\), relative to a fixed origin \(O\), is given by $$\mathbf{v} = (1 - 2t)\mathbf{i} + (2t^2 + t - 13)\mathbf{j}.$$

1. Show that \(P\) is never stationary. [2]
2. Find, in terms of \(\mathbf{i}\) and \(\mathbf{j}\), the acceleration of \(P\) at time \(t\). [1]

The mass of \(P\) is 0.5 kg.

1. Determine the magnitude of \(\mathbf{F}\) when \(P\) is moving in the direction of the vector \(-2\mathbf{i} + \mathbf{j}\). Give your answer correct to 3 significant figures. [5]

When \(t = 1\), \(P\) is at the point with position vector \(\frac{1}{6}\mathbf{j}\).

1. Determine the bearing of \(P\) from \(O\) at time \(t = 1.5\). [5]

A remote-controlled toy car is moving over a horizontal surface. It moves in a straight line through a point \(A\). The toy is initially at the point with displacement \(3\) metres from \(A\). Its velocity, \(v\,\mathrm{m}\,\mathrm{s}^{-1}\), at time \(t\) seconds is defined by $$v = 0.06(2 + t - t^2)$$

1. Find an expression for the displacement, \(r\) metres, of the toy from \(A\) at time \(t\) seconds. [4 marks]
2. In this question use \(g = 9.8\,\mathrm{m}\,\mathrm{s}^{-2}\) At time \(t = 2\) seconds, the toy launches a ball which travels directly upwards with initial speed \(3.43\,\mathrm{m}\,\mathrm{s}^{-1}\) Find the time taken for the ball to reach its highest point. [3 marks]

A car, starting from rest, is driven along a horizontal track. The velocity of the car, \(v \text{m s}^{-1}\), at time \(t\) seconds, is modelled by the equation $$v = 0.48t^2 - 0.024t^3 \text{ for } 0 \leq t \leq 15$$

1. Find the distance the car travels during the first 10 seconds of its journey. [3 marks]
2. Find the maximum speed of the car. Give your answer to three significant figures. [4 marks]
3. Deduce the range of values of \(t\) for which the car is modelled as decelerating. [2 marks]

A particle, \(P\), is moving in a straight line with acceleration \(a\text{ m s}^{-2}\) at time \(t\) seconds, where $$a = 4 - 3t^2$$

1. Initially \(P\) is stationary. Find an expression for the velocity of \(P\) in terms of \(t\). [2 marks]
2. When \(t = 2\), the displacement of \(P\) from a fixed point, O, is 39 metres. Find the time at which \(P\) passes through O, giving your answer to three significant figures. Fully justify your answer. [5 marks]

A particle, P, is moving along a straight line such that its acceleration \(a\) m s⁻², at any time, \(t\) seconds, may be modelled by $$a = 3 + 0.2t$$ When \(t = 2\), the velocity of P is \(k\) m s⁻¹

1. Show that the initial velocity of P is given by the expression \((k - 6.4)\) m s⁻¹ [4 marks]
2. The initial velocity of P is one fifth of the velocity when \(t = 2\) Find the value of \(k\). [2 marks]

\includegraphics{figure_17} A car and caravan, connected by a tow bar, move forward together along a horizontal road. Their velocity \(v\) m s\(^{-1}\) at time \(t\) seconds, for \(0 \leq t < 20\), is given by $$v = 0.5t + 0.01t^2$$

1. Show that when \(t = 15\) their acceleration is 0.8 m s\(^{-2}\) [2 marks]
2. The car has a mass of 1500 kg The caravan has a mass of 850 kg When \(t = 15\) the tension in the tow bar is 800 N and the car experiences a resistance force of 100 N
   1. Find the total resistance force experienced by the caravan when \(t = 15\) [2 marks]
   2. Find the driving force being applied by the car when \(t = 15\) [3 marks]
3. State one assumption you have made about the tow bar. [1 mark]

A toy remote control speed boat is launched from one edge of a small pond and moves in a straight line across the pond's surface. The boat's velocity, \(v \text{ m s}^{-1}\), is modelled in terms of time, \(t\) seconds after the boat is launched, by the expression $$v = 0.9 + 0.16t - 0.06t^2$$

1. Find the acceleration of the boat when \(t = 2\) [3 marks]
2. Find the displacement of the boat, from the point where it was launched, when \(t = 2\) [4 marks]

A particle moves in a straight line with acceleration \(a\) m s\(^{-2}\), at time \(t\) seconds, where $$a = 10 - 6t$$ The particle's velocity, \(v\) m s\(^{-1}\), and displacement, \(r\) metres, are both initially zero. Show that $$r = t^2(5 - t)$$ Fully justify your answer. [4 marks]