3.02d Constant acceleration: SUVAT formulae

A train \(T_1\) moves from rest at Station \(A\) with constant acceleration \(2 \text{ m s}^{-2}\) until it reaches a speed of \(36 \text{ m s}^{-1}\). In maintains this constant speed for 90 s before the brakes are applied, which produce constant retardation \(3 \text{ m s}^{-2}\). The train \(T_1\) comes to rest at station \(B\).

1. Sketch a speed-time graph to illustrate the journey of \(T_1\) from \(A\) to \(B\). [3]
2. Show that the distance between \(A\) and \(B\) is 3780 m. [5]

\includegraphics{figure_3} A second train \(T_2\) takes 150 s to move form rest at \(A\) to rest at \(B\). Figure 3 shows the speed-time graph illustrating this journey.

1. Explain briefly one way in which \(T_1\)'s journey differs from \(T_2\)'s journey. [1]
2. Find the greatest speed, in m s\(^{-1}\), attained by \(T_2\) during its journey. [3]

\includegraphics{figure_3} The object of a game is to throw a ball \(B\) from a point \(A\) to hit a target \(T\) which is placed at the top of a vertical pole, as shown in Figure 3. The point \(A\) is 1 m above horizontal ground and the height of the pole is 2 m. The pole is at a horizontal distance of 10 m from \(A\). The ball \(B\) is projected from \(A\) with a speed of 11 m s\(^{-1}\) at an angle of elevation of \(30°\). The ball hits the pole at the point \(C\). The ball \(B\) and the target \(T\) are modelled as particles.

1. Calculate, to 2 decimal places, the time taken for \(B\) to move from \(A\) to \(C\). [3]
2. Show that \(C\) is approximately 0.63 m below \(T\). [4]

The ball is thrown again from \(A\). The speed of projection of \(B\) is increased to \(V\) m s\(^{-1}\), the angle of elevation remaining \(30°\). This time \(B\) hits \(T\).

1. Calculate the value of \(V\). [6]
2. Explain why, in practice, a range of values of \(V\) would result in \(B\) hitting the target. [1]

\includegraphics{figure_3} A particle \(P\) is projected from a point \(A\) with speed \(u\) m s\(^{-1}\) at an angle of elevation \(\theta\), where \(\cos \theta = \frac{4}{5}\). The point \(B\), on horizontal ground, is vertically below \(A\) and \(AB = 45\) m. After projection, \(P\) moves freely under gravity passing through a point \(C\), 30 m above the ground, before striking the ground at the point \(D\), as shown in Figure 3. Given that \(P\) passes through \(C\) with speed 24.5 m s\(^{-1}\),

1. using conservation of energy, or otherwise, show that \(u = 17.5\), [4]
2. find the size of the angle which the velocity of \(P\) makes with the horizontal as \(P\) passes through \(C\), [3]
3. find the distance \(BD\). [7]

\includegraphics{figure_3} [In this question, the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are in a vertical plane, \(\mathbf{i}\) being horizontal and \(\mathbf{j}\) being vertical.] A particle \(P\) is projected from the point \(A\) which has position vector \(47.5\mathbf{j}\) metres with respect to a fixed origin \(O\). The velocity of projection of \(P\) is \((2u\mathbf{i} + 5u\mathbf{j})\) m s\(^{-1}\). The particle moves freely under gravity passing through the point \(B\) with position vector \(30\mathbf{i}\) metres, as shown in Figure 3.

1. Show that the time taken for \(P\) to move from \(A\) to \(B\) is 5 s. [6]
2. Find the value of \(u\). [2]
3. Find the speed of \(P\) at \(B\). [5]

\includegraphics{figure_2} At time \(t = 0\) a small package is projected from a point \(B\) which is 2.4 m above a point \(A\) on horizontal ground. The package is projected with speed 23.75 m s\(^{-1}\) at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac{4}{3}\). The package strikes the ground at the point \(C\), as shown in Fig. 2. The package is modelled as a particle moving freely under gravity.

1. Find the time taken for the package to reach \(C\). [5]

A lorry moves along the line \(AC\), approaching \(A\) with constant speed 18 m s\(^{-1}\). At time \(t = 0\) the rear of the lorry passes \(A\) and the lorry starts to slow down. It comes to rest \(T\) seconds later. The acceleration, \(a\) m s\(^{-2}\) of the lorry at time \(t\) seconds is given by $$a = -\frac{1}{4}t^2, \quad 0 \leq t \leq T.$$

1. Find the speed of the lorry at time \(t\) seconds. [3]
2. Hence show that \(T = 6\). [3]
3. Show that when the package reaches \(C\) it is just under 10 m behind the rear of the moving lorry. [5]

END

A particle is projected from a point with speed \(u\) at an angle of elevation \(\alpha\) above the horizontal and moves freely under gravity. When it has moved a horizontal distance \(x\), its height above the point of projection is \(y\).

1. Show that $$y = x \tan \alpha - \frac{gx^2}{2u^2}(1 + \tan^2 \alpha).$$ [5]

A shot-putter puts a shot from a point \(A\) at a height of 2 m above horizontal ground. The shot is projected at an angle of elevation of 45° with a speed of 14 m s\(^{-1}\). By modelling the shot as a particle moving freely under gravity,

1. find, to 3 significant figures, the horizontal distance of the shot from \(A\) when the shot hits the ground, [5]
2. find, to 2 significant figures, the time taken by the shot in moving from \(A\) to reach the ground. [2]

\includegraphics{figure_3} A ball is thrown from a point 4 m above horizontal ground. The ball is projected at an angle \(\alpha\) above the horizontal, where \(\tan \alpha = \frac{4}{3}\). The ball hits the ground at a point which is a horizontal distance 8 m from its point of projection, as shown in Fig. 3. The initial speed of the ball is \(u\) m s\(^{-1}\) and the time of flight is \(T\) seconds.

1. Prove that \(uT = 10\). [2]
2. Find the value of \(u\). [5]

As the ball hits the ground, its direction of motion makes an angle \(\phi\) with the horizontal.

1. Find \(\tan \phi\). [5]

A vertical cliff is 73.5 m high. Two stones \(A\) and \(B\) are projected simultaneously. Stone \(A\) is projected horizontally from the top of the cliff with speed 28 m s\(^{-1}\). Stone \(B\) is projected from the bottom of the cliff with speed 35 m s\(^{-1}\) at an angle \(\alpha\) above the horizontal. The stones move freely under gravity in the same vertical plane and collide in mid-air. By considering the horizontal motion of each stone,

1. prove that \(\cos \alpha = \frac{4}{5}\). [4]

1. Find the time which elapses between the instant when the stones are projected and the instant when they collide. [4]

\includegraphics{figure_3} A ball is projected with speed 40 m s\(^{-1}\) from a point \(P\) on a cliff above horizontal ground. The point \(O\) on the ground is vertically below \(P\) and \(OP\) is 36 m. The ball is projected at an angle \(\theta°\) to the horizontal. The point \(Q\) is the highest point of the path of the ball and is 12 m above the level of \(P\). The ball moves freely under gravity and hits the ground at the point \(R\), as shown in Figure 3. Find

1. the value of \(\theta\), (3)
2. the distance \(OR\), (6)
3. the speed of the ball as it hits the ground at \(R\). (3)

A particle is projected from a point \(O\) with speed \(u\) at an angle of elevation \(\alpha\) above the horizontal and moves freely under gravity. When the particle has moved a horizontal distance \(x\), its height above \(O\) is \(y\).

1. Show that $$y = x \tan \alpha - \frac{gx^2}{2u^2 \cos^2 \alpha}$$ [4]

A girl throws a ball from a point \(A\) at the top of a cliff. The point \(A\) is 8 m above a horizontal beach. The ball is projected with speed 7 m s\(^{-1}\) at an angle of elevation of 45°. By modelling the ball as a particle moving freely under gravity,

1. find the horizontal distance of the ball from \(A\) when the ball is 1 m above the beach. [5]

A boy is standing on the beach at the point \(B\) vertically below \(A\). He starts to run in a straight line with speed \(v\) m s\(^{-1}\), leaving \(B\) 0.4 seconds after the ball is thrown. He catches the ball when it is 1 m above the beach.

1. Find the value of \(v\). [4]

A particle \(P\) moves in a straight line through a fixed point \(O\) with constant acceleration \(a\) ms\(^{-2}\). 3 seconds after passing through \(O\), \(P\) is 6 m from \(O\). After a further 6 seconds, \(P\) has travelled a further 33 m in the same direction. Calculate

1. the value of \(a\), [5 marks]
2. the speed with which \(P\) passed through \(O\). [2 marks]

A car starts from rest at time \(t = 0\) and moves along a straight road with constant acceleration 4 ms\(^{-2}\) for 10 seconds. It then travels at a constant speed for 50 seconds before decelerating to rest over a further distance of 240 m.

1. Sketch a graph of velocity against time for the total period of the car's motion. [3 marks]
2. Find the car's average speed for the whole journey. [6 marks]

In reality the car's acceleration \(a\) ms\(^{-2}\) in the first 10 seconds is not constant, but increases from 0 to 4 ms\(^{-2}\) in the first 5 seconds and then decreases to 0 again. A refined model designed to take account of this uses the formula \(a = k(mt - t^2)\) for \(0 \leq t \leq 10\).

1. Calculate the values of the constants \(k\) and \(m\). [5 marks]
2. Find the acceleration of the car when \(t = 2\) according to this model. [2 marks]

A car moves in a straight line from \(P\) to \(Q\), a distance of \(420\) m, with constant acceleration. At \(P\) the speed of the car is \(8\) ms\(^{-1}\). At \(Q\) the speed of the car is \(20\) ms\(^{-1}\). Find

1. the time taken to travel from \(P\) to \(Q\), \hfill [2 marks]
2. the acceleration of the car, \hfill [2 marks]
3. the time taken for the car to travel \(240\) m from \(P\). \hfill [4 marks]

Given that the mass of the car is \(1200\) kg and the tractive force of the car is \(900\) N,

1. find the magnitude of the resistance to the car's motion. \hfill [3 marks]

The diagram shows the velocity-time graph for a cyclist's journey. Each section has constant acceleration or deceleration and the three sections are of equal duration \(x\) seconds each. \includegraphics{figure_6} Given that the total distance travelled is \(792\) m,

1. find the value of \(x\) and the acceleration for the first section of the journey. [6 marks]

Another cyclist covers the same journey in three sections of equal duration, accelerating at \(\frac{1}{11} \text{ ms}^{-2}\) for the first section, travelling at constant speed for the second section and decelerating at \(\frac{1}{11} \text{ ms}^{-2}\) for the third section.

1. Find the time taken by this cyclist to complete the journey. [6 marks]
2. Show that the maximum speeds of both cyclists are the same. [2 marks]

A train starts from rest at a station \(S\) and accelerates at a constant rate for \(2x\) seconds to a speed of \(5x\) ms\(^{-1}\). It maintains this speed until 126 seconds after it left \(S\) and then decelerates at a constant rate until it comes to rest at another station \(T\), \(20x\) seconds after it left \(S\).

1. Sketch a velocity-time graph for this journey. [4 marks]

Given that the distance between \(S\) and \(T\) is \(5.4\) km,

1. show that \(x^2 + 7x = 120\). [4 marks]
2. Find the value of \(x\). [3 marks]

The velocity-time graph illustrates the motion of a particle which accelerates from rest to 8 ms\(^{-1}\) in \(x\) seconds and then to 24 ms\(^{-1}\) in a further 4 seconds. It then travels at a constant speed for another \(y\) seconds before decelerating to 12 ms\(^{-1}\) over the next \(y\) seconds and then to rest in the final 7 seconds of its motion. \includegraphics{figure_6} Given that the total distance travelled by the particle is 496 m,

1. show that \(2x + 21y = 195\). [4 marks]

Given also that the average speed of the particle during its motion is 15.5 ms\(^{-1}\),

1. show that \(x + 2y = 21\). [3 marks]
2. Hence find the values of \(x\) and \(y\). [3 marks]
3. Write down the acceleration for each section of the motion. [3 marks]

A body moves in a straight line with constant acceleration. Its speed increases from 17 ms\(^{-1}\) to 33 ms\(^{-1}\) while it travels a distance of 250 m. Find

1. the time taken to travel the 250 m, \hfill [3 marks]
2. the acceleration of the body. \hfill [2 marks]

The body now decelerates at a constant rate from 33 ms\(^{-1}\) to rest in 6 seconds.

1. Find the distance travelled in these 6 seconds. \hfill [2 marks]

Two small spheres \(P\) and \(Q\) have masses \(0.1\) kg and \(0.2\) kg respectively. The spheres are moving directly towards each other on a horizontal plane and collide. Immediately before the collision \(P\) has speed \(4\) m s\(^{-1}\) and \(Q\) has speed \(3\) m s\(^{-1}\). Immediately after the collision the spheres move away from each other, \(P\) with speed \(u\) m s\(^{-1}\) and \(Q\) with speed \((3.5 - u)\) m s\(^{-1}\).

1. Find the value of \(u\). [4]

After the collision the spheres both move with deceleration of magnitude \(5\) m s\(^{-2}\) until they come to rest on the plane.

1. Find the distance \(PQ\) when both \(P\) and \(Q\) are at rest. [4]

A particle moves downwards on a smooth plane inclined at an angle \(\alpha\) to the horizontal. The particle passes through the point \(P\) with speed \(u\) m s\(^{-1}\). The particle travels \(2\) m during the first \(0.8\) s after passing through \(P\), then a further \(6\) m in the next \(1.2\) s. Find

1. the value of \(u\) and the acceleration of the particle, [7]
2. the value of \(\alpha\) in degrees. [2]

\includegraphics{figure_1} Particles \(P\) and \(Q\), of masses \(0.3\) kg and \(0.4\) kg respectively, are attached to the ends of a light inextensible string. The string passes over a smooth fixed pulley. The system is in motion with the string taut and with each of the particles moving vertically. The downward acceleration of \(P\) is \(a\) m s\(^{-2}\) (see diagram).

1. Show that \(a = -1.4\). [4]

Initially \(P\) and \(Q\) are at the same horizontal level. \(P\)'s initial velocity is vertically downwards and has magnitude \(2.8\) m s\(^{-1}\).

1. Assuming that \(P\) does not reach the floor and that \(Q\) does not reach the pulley, find the time taken for \(P\) to return to its initial position. [3]

A man drives a car on a horizontal straight road. At \(t = 0\), where the time \(t\) is in seconds, the car runs out of petrol. At this instant the car is moving at \(12\) m s\(^{-1}\). The car decelerates uniformly, coming to rest when \(t = 8\). The man then walks back along the road at \(0.7\) m s\(^{-1}\) until he reaches a petrol station a distance of \(420\) m from his car. After his arrival at the petrol station it takes him \(250\) s to obtain a can of petrol. He is then given a lift back to his car on a motorcycle. The motorcycle starts from rest and accelerates uniformly until its speed is \(20\) m s\(^{-1}\); it then decelerates uniformly, coming to rest at the stationary car at time \(t = T\).

1. Sketch the shape of the \((t, v)\) graph for the man for \(0 \leq t \leq T\). [Your sketch need not be drawn to scale; numerical values need not be shown.] [5]
2. Find the deceleration of the car for \(0 < t < 8\). [2]
3. Find the value of \(T\). [4]

\includegraphics{figure_7} \(PQ\) is a line of greatest slope, of length \(4\) m, on a smooth plane inclined at \(30°\) to the horizontal. Particles \(A\) and \(B\), of masses \(0.15\) kg and \(0.5\) kg respectively, move along \(PQ\) with \(A\) below \(B\). The particles are both moving upwards, \(A\) with speed \(8\) m s\(^{-1}\) and \(B\) with speed \(2\) m s\(^{-1}\), when they collide at the mid-point of \(PQ\) (see diagram). Particle \(A\) is instantaneously at rest immediately after the collision.

1. Show that \(B\) does not reach \(Q\) in the subsequent motion. [8]
2. Find the time interval between the instant of \(A\)'s arrival at \(P\) and the instant of \(B\)'s arrival at \(P\). [6]

A particle of mass \(0.1\) kg is at rest at a point \(A\) on a rough plane inclined at \(15°\) to the horizontal. The particle is given an initial velocity of \(6\) m s\(^{-1}\) and starts to move up a line of greatest slope of the plane. The particle comes to instantaneous rest after \(1.5\) s.

1. Find the coefficient of friction between the particle and the plane. [7]
2. Show that, after coming to instantaneous rest, the particle moves down the plane. [2]
3. Find the speed with which the particle passes through \(A\) during its downward motion. [6]

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 \(1.8t\) m s\(^{-2}\), and for \(4 \leq t \leq 7\) the particle has constant acceleration \(7.2\) 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_6} The diagram shows the \((t, v)\) graph for the motion of a hoist used to deliver materials to different levels at a building site. The hoist moves vertically. The graph consists of straight line segments. In the first stage the hoist travels upwards from ground level for \(25\) s, coming to rest \(8\) m above ground level.

1. Find the greatest speed reached by the hoist during this stage. [2]

The second stage consists of a \(40\) s wait at the level reached during the first stage. In the third stage the hoist continues upwards until it comes to rest \(40\) m above ground level, arriving \(135\) s after leaving ground level. The hoist accelerates at \(0.02\) m s\(^{-2}\) for the first \(40\) s of the third stage, reaching a speed of \(V\) m s\(^{-1}\). Find

1. the value of \(V\), [3]
2. the length of time during the third stage for which the hoist is moving at constant speed, [4]
3. the deceleration of the hoist in the final part of the third stage. [3]