3.02d Constant acceleration: SUVAT formulae

A bus travels between two stops, \(A\) and \(B\). The bus starts from rest at \(A\) and accelerates at a constant rate of \(a \text{ ms}^{-2}\) until it reaches a speed of \(16 \text{ ms}^{-1}\). It then travels at this constant speed before decelerating at a constant rate of \(0.75 \text{ ms}^{-2}\), coming to rest at \(B\). The total time for the journey is \(240\) s.

1. Sketch the velocity-time graph for the bus's journey from \(A\) to \(B\). [1]
2. Find an expression, in terms of \(a\), for the length of time that the bus is travelling with constant speed. [2]
3. Given that the distance from \(A\) to \(B\) is \(3000\) m, find the value of \(a\). [3]

A particle moves in a straight line. It starts from rest, at time \(t = 0\), and accelerates at 0.6 t ms\(^{-2}\) for 4 s, reaching a speed of \(V\) ms\(^{-1}\). The particle then travels at \(V\) ms\(^{-1}\) for 11 s, and finally slows down, with constant deceleration, stopping after a further 5 s.

1. Show that \(V = 4.8\). [1]
2. Sketch a velocity-time graph for the motion. [3]
3. Find an expression, in terms of \(t\), for the velocity of the particle for \(15 \leqslant t \leqslant 20\). [2]
4. Find the total distance travelled by the particle. [4]

\(A\) and \(B\) are points on the same line of greatest slope of a rough plane inclined at \(30°\) to the horizontal. \(A\) is higher up the plane than \(B\) and the distance \(AB\) is \(2.25 \text{ m}\). A particle \(P\), of mass \(m \text{ kg}\), is released from rest at \(A\) and reaches \(B\) \(1.5 \text{ s}\) later. Find the coefficient of friction between \(P\) and the plane. [6]

\includegraphics{figure_6} The diagram shows the velocity-time graph for a lift moving between floors in a building. The graph consists of straight line segments. In the first stage the lift travels downwards from the ground floor for \(5 \text{ s}\), coming to rest at the basement after travelling \(10 \text{ m}\).

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

The second stage consists of a \(10 \text{ s}\) wait at the basement. In the third stage, the lift travels upwards until it comes to rest at a floor \(34.5 \text{ m}\) above the basement, arriving \(24.5 \text{ s}\) after the start of the first stage. The lift accelerates at \(2 \text{ m s}^{-2}\) for the first \(3 \text{ s}\) of the third stage, reaching a speed of \(V \text{ m s}^{-1}\). Find

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

\includegraphics{figure_6} Particles \(A\) and \(B\) are attached to the ends of a light inextensible string which passes over a smooth pulley. The system is held at rest with the string taut and its straight parts vertical. Both particles are at a height of 0.36 m above the floor (see diagram). The system is released and \(A\) begins to fall, reaching the floor after 0.6 s.

1. Find the acceleration of \(A\) as it falls. [2]

The mass of \(A\) is 0.45 kg. Find

1. the tension in the string while \(A\) is falling, [2]
2. the mass of \(B\), [3]
3. the maximum height above the floor reached by \(B\). [3]

A particle \(P\) travels in a straight line from \(A\) to \(D\), passing through the points \(B\) and \(C\). For the section \(AB\) the velocity of the particle is \((0.5t - 0.01t^2)\) m s\(^{-1}\), where \(t\) s is the time after leaving \(A\).

1. Given that the acceleration of \(P\) at \(B\) is 0.1 m s\(^{-2}\), find the time taken for \(P\) to travel from \(A\) to \(B\). [3]

The acceleration of \(P\) from \(B\) to \(C\) is constant and equal to 0.1 m s\(^{-2}\).

1. Given that \(P\) reaches \(C\) with speed 14 m s\(^{-1}\), find the time taken for \(P\) to travel from \(B\) to \(C\). [3]

\(P\) travels with constant deceleration 0.3 m s\(^{-2}\) from \(C\) to \(D\). Given that the distance \(CD\) is 300 m, find

1. the speed with which \(P\) reaches \(D\), [2]
2. the distance \(AD\). [6]

\includegraphics{figure_6} Particles \(A\) and \(B\), of masses 0.2 kg and 0.45 kg respectively, are connected by a light inextensible string of length 2.8 m. The string passes over a small smooth pulley at the edge of a rough horizontal surface, which is 2 m above the floor. Particle \(A\) is held in contact with the surface at a distance of 2.1 m from the pulley and particle \(B\) hangs freely (see diagram). The coefficient of friction between \(A\) and the surface is 0.3. Particle \(A\) is released and the system begins to move.

1. Find the acceleration of the particles and show that the speed of \(B\) immediately before it hits the floor is 3.95 m s\(^{-1}\), correct to 3 significant figures. [7]
2. Given that \(B\) remains on the floor, find the speed with which \(A\) reaches the pulley. [4]

\includegraphics{figure_6} Particles \(A\) and \(B\), of masses \(0.2 \text{ kg}\) and \(0.45 \text{ kg}\) respectively, are connected by a light inextensible string of length \(2.8 \text{ m}\). The string passes over a small smooth pulley at the edge of a rough horizontal surface, which is \(2 \text{ m}\) above the floor. Particle \(A\) is held in contact with the surface at a distance of \(2.1 \text{ m}\) from the pulley and particle \(B\) hangs freely (see diagram). The coefficient of friction between \(A\) and the surface is \(0.3\). Particle \(A\) is released and the system begins to move.

1. Find the acceleration of the particles and show that the speed of \(B\) immediately before it hits the floor is \(3.95 \text{ m s}^{-1}\), correct to 3 significant figures. [7]
2. Given that \(B\) remains on the floor, find the speed with which \(A\) reaches the pulley. [4]

A car has mass \(1250 \text{ kg}\).

1. The car is moving along a straight level road at a constant speed of \(36 \text{ m s}^{-1}\) and is subject to a constant resistance of magnitude \(850 \text{ N}\). Find, in kW, the rate at which the engine of the car is working. [2]
2. The car travels at a constant speed up a hill and is subject to the same resistance as in part (i). The hill is inclined at an angle of \(\theta°\) to the horizontal, where \(\sin \theta° = 0.1\), and the engine is working at \(63 \text{ kW}\). Find the speed of the car. [3]
3. The car descends the same hill with the engine of the car working at a constant rate of \(20 \text{ kW}\). The resistance is not constant. The initial speed of the car is \(20 \text{ m s}^{-1}\). Eight seconds later the car has speed \(24 \text{ m s}^{-1}\) and has moved \(176 \text{ m}\) down the hill. Use an energy method to find the total work done against the resistance during the eight seconds. [5]

\includegraphics{figure_1} The diagram shows the velocity-time graph for a train which travels from rest at one station to rest at the next station. The graph consists of three straight line segments. The distance between the two stations is 9040 m.

1. Find the acceleration of the train during the first 40 s. [1]
2. Find the length of time for which the train is travelling at constant speed. [2]
3. Find the distance travelled by the train while it is decelerating. [2]

A small ball is projected vertically downwards with speed \(5\text{ m s}^{-1}\) from a point \(A\) at a height of \(7.2\text{ m}\) above horizontal ground. The ball hits the ground with speed \(V\text{ m s}^{-1}\) and rebounds vertically upwards with speed \(\frac{1}{2}V\text{ m s}^{-1}\). The highest point the ball reaches after rebounding is \(B\). Find \(V\) and hence find the total time taken for the ball to reach the ground from \(A\) and rebound to \(B\). [5]

A car moves in a straight line with initial speed \(u\) m s\(^{-1}\) and constant acceleration \(a\) m s\(^{-2}\). The car takes 5 s to travel the first 80 m and it takes 8 s to travel the first 160 m. Find \(a\) and \(u\). [6]

A particle is projected vertically upwards with speed \(30\) m s\(^{-1}\) from a point on horizontal ground.

1. Show that the maximum height above the ground reached by the particle is \(45\) m. [2]
2. Find the time that it takes for the particle to reach a height of \(33.75\) m above the ground for the first time. Find also the speed of the particle at this time. [4]

\includegraphics{figure_5} The velocity of a particle moving in a straight line is \(v\) m s\(^{-1}\) at time \(t\) seconds after leaving a fixed point \(O\). The diagram shows a velocity-time graph which models the motion of the particle from \(t = 0\) to \(t = 16\). The graph consists of five straight line segments. The acceleration of the particle from \(t = 0\) to \(t = 3\) is \(\frac{7}{3}\) m s\(^{-2}\). The velocity of the particle at \(t = 5\) is \(7\) m s\(^{-1}\) and it comes to instantaneous rest at \(t = 8\). The particle then comes to rest again at \(t = 16\). The minimum velocity of the particle is \(V\) m s\(^{-1}\).

1. Find the distance travelled by the particle in the first \(8\) s of its motion. [3]
2. Given that when the particle comes to rest at \(t = 16\) its displacement from \(O\) is \(32\) m, find the value of \(V\). [4]

A particle is projected vertically upwards from a point \(O\) with initial speed \(12.5 \text{ m s}^{-1}\). At the same instant another particle is released from rest at a point 10 m vertically above \(O\). Find the height above \(O\) at which the particles meet. [5]

A car travels along a straight road with constant acceleration. It passes through points \(A\), \(B\) and \(C\). The car passes point \(A\) with velocity 14 m s\(^{-1}\). The two sections \(AB\) and \(BC\) are of equal length. The times taken to travel along \(AB\) and \(BC\) are 5 s and 3 s respectively.

1. Write down an expression for the distance \(AB\) in terms of the acceleration of the car. Write down a similar expression for the distance \(AC\). Hence show that the acceleration of the car is 4 m s\(^{-2}\). [4]
2. Find the speed of the car as it passes point \(C\). [2]

A particle \(P\) is projected vertically upwards from horizontal ground with speed 12 m s\(^{-1}\).

1. Find the time taken for \(P\) to return to the ground. [2]

The time in seconds after \(P\) is projected is denoted by \(t\). When \(t = 1\), a second particle \(Q\) is projected vertically upwards with speed 10 m s\(^{-1}\) from a point which is 5 m above the ground. Particles \(P\) and \(Q\) move in different vertical lines.

1. Find the set of values of \(t\) for which the two particles are moving in the same direction. [4]

\includegraphics{figure_3} The velocity of a particle moving in a straight line is \(v\) m s\(^{-1}\) at time \(t\) seconds. The diagram shows a velocity-time graph which models the motion of the particle from \(t = 0\) to \(t = T\). The graph consists of four straight line segments. The particle reaches its maximum velocity \(V\) m s\(^{-1}\) at \(t = 10\).

1. Find the acceleration of the particle during the first \(2\) seconds. [1]
2. Find the value of \(V\). [2]

At \(t = 6\), the particle is instantaneously at rest at the point \(A\). At \(t = T\), the particle comes to rest at the point \(B\). At \(t = 0\) the particle starts from rest at a point one third of the way from \(A\) to \(B\).

1. Find the distance \(AB\) and hence find the value of \(T\). [4]

A particle of mass \(0.3\) kg is released from rest above a tank containing water. The particle falls vertically, taking \(0.8\) s to reach the water surface. There is no instantaneous change of speed when the particle enters the water. The depth of water in the tank is \(1.25\) m. The water exerts a force on the particle resisting its motion. The work done against this resistance force from the instant that the particle enters the water until it reaches the bottom of the tank is \(1.2\) J.

1. Use an energy method to find the speed of the particle when it reaches the bottom of the tank. [4]

When the particle reaches the bottom of the tank, it bounces back vertically upwards with initial speed \(7\) m s\(^{-1}\). As the particle rises through the water, it experiences a constant resistance force of \(1.8\) N. The particle comes to instantaneous rest \(t\) seconds after it bounces on the bottom of the tank.

1. Find the value of \(t\). [7]

\includegraphics{figure_4} Two blocks \(A\) and \(B\) of masses 4 kg and 5 kg respectively are joined by a light inextensible string. The blocks rest on a smooth plane inclined at an angle \(\alpha\) to the horizontal, where tan \(\alpha = \frac{7}{24}\). The string is parallel to a line of greatest slope of the plane with \(B\) above \(A\). A force of magnitude 36 N acts on \(B\), parallel to a line of greatest slope of the plane (see diagram).

1. Find the acceleration of the blocks and the tension in the string. [5]

1. At a particular instant, the speed of the blocks is 1 m s\(^{-1}\). Find the time, after this instant, that it takes for the blocks to travel 0.65 m. [2]

\includegraphics{figure_2} The diagram shows a velocity-time graph which models the motion of a tractor. The graph consists of four straight line segments. The tractor passes a point \(O\) at time \(t = 0\) with speed \(U\) m s\(^{-1}\). The tractor accelerates to a speed of \(V\) m s\(^{-1}\) over a period of 5 s, and then travels at this speed for a further 25 s. The tractor then accelerates to a speed of 12 m s\(^{-1}\) over a period of 5 s. The tractor then decelerates to rest over a period of 15 s.

1. Given that the acceleration of the tractor between \(t = 30\) and \(t = 35\) is 0.8 m s\(^{-2}\), find the value of \(V\). [2]
2. Given also that the total distance covered by the tractor in the 50 seconds of motion is 375 m, find the value of \(U\). [3]

Two particles \(A\) and \(B\) move in the same vertical line. Particle \(A\) is projected vertically upwards from the ground with speed 20 m s\(^{-1}\). One second later particle \(B\) is dropped from rest from a height of 40 m.

1. Find the height above the ground at which the two particles collide. [4]
2. Find the difference in the speeds of the two particles at the instant when the collision occurs. [3]

A block of mass 3 kg is initially at rest on a rough horizontal plane. A force of magnitude 6 N is applied to the block at an angle of \(\theta\) above the horizontal, where \(\cos \theta = \frac{24}{25}\). The force is applied for a period of 5 s, during which time the block moves a distance of 4.5 m.

1. Find the magnitude of the frictional force on the block. [4]
2. Show that the coefficient of friction between the block and the plane is 0.165, correct to 3 significant figures. [3]
3. When the block has moved a distance of 4.5 m, the force of magnitude 6 N is removed and the block then decelerates to rest. Find the total time for which the block is in motion. [4]

\includegraphics{figure_7} Two particles \(P\) and \(Q\), of masses 0.3 kg and 0.2 kg respectively, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley which is attached to the edge of a smooth plane. The plane is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac{3}{5}\). \(P\) lies on the plane and \(Q\) hangs vertically below the pulley at a height of 0.8 m above the floor (see diagram). The string between \(P\) and the pulley is parallel to a line of greatest slope of the plane. \(P\) is released from rest and \(Q\) moves vertically downwards.

1. Find the tension in the string and the magnitude of the acceleration of the particles. [5]

\(Q\) hits the floor and does not bounce. It is given that \(P\) does not reach the pulley in the subsequent motion.

1. Find the time, from the instant at which \(P\) is released, for \(Q\) to reach the floor. [2]
2. When \(Q\) hits the floor the string becomes slack. Find the time, from the instant at which \(P\) is released, for the string to become taut again. [4]