Kinematics in a straight line

A bus is travelling along a straight horizontal road at 16 m s⁻¹ when the driver applies the brakes. The bus decelerates uniformly and comes to rest after travelling 64 m. The bus is modelled as a particle.

A coin is dropped from rest from a footbridge and falls vertically into the river below. It reaches the water 2.5 s after it is released. The particle model is used, air resistance is ignored and g = 9.8 m s⁻².

A jogger runs along a straight path that starts at her front gate H. Her displacement–time graph consists of three straight line segments joining the points (0, 0), (300, 900), (420, 900) and (720, 0), where t is the time in seconds and s is her displacement from H in metres.

A speed skater moves along a straight section of an ice track with constant acceleration 0.8 m s⁻². During a period of 6.0 s her speed increases to 11.4 m s⁻¹. The skater is modelled as a particle.

A lift moves vertically upwards from rest at the ground floor of a building and comes to rest at a higher floor. Figure 1 shows the velocity–time graph of the lift for the whole journey. The lift is modelled as a particle.

A robot cleaner moves along a straight corridor. Figure 1 shows the graph of its displacement s metres from a fixed point O in the corridor against time t seconds, for 0 ≤ t ≤ 20. The graph consists of four straight line segments. The robot is modelled as a particle.

Figure 1 shows a ball thrown vertically downwards with speed 3.5 m s⁻¹ from a window W. The ball falls past the level of a balcony B, without touching it, and hits the horizontal ground. The heights of W and B above the ground are shown in the figure. The particle model is used, air resistance is ignored and g = 9.8 m s⁻².

A tram moves along the straight track shown in Figure 1. It passes point A and accelerates uniformly until it passes point B. It then decelerates uniformly from B and comes to rest at point C. Its speeds at A and B and the distances AB and BC are shown in the figure. The tram is modelled as a particle.

A pebble is fired vertically upwards from a catapult at ground level with speed 17.5 m s⁻¹. The particle model is used, air resistance is ignored and g = 9.8 m s⁻².

A camera drone rises vertically from rest on the ground. It accelerates uniformly for 4 s until its speed is 6 m s⁻¹. It then rises at a constant speed of 6 m s⁻¹ for T seconds, and then decelerates uniformly, coming to rest after a further 3 s. It is then 90 m above the ground. The drone is modelled as a particle.

Two cyclists, A and B, ride towards each other along a straight cycle path. At time t = 0 they are 200 m apart. A starts from rest and accelerates uniformly at 0.4 m s⁻². B rides at a constant speed of 4 m s⁻¹ throughout. The cyclists are modelled as particles and A is still accelerating when they meet.

An automated cart moves along a straight track in a warehouse. At time t = 0 its velocity is 0.5 m s⁻¹. Its acceleration–time graph shows that its acceleration is 0.25 m s⁻² for 0 < t < 6, zero for 6 < t < 26, and −0.4 m s⁻² from t = 26 until the cart comes to rest at t = T. The cart is modelled as a particle.

A hot-air balloon is rising vertically with constant speed 4.9 m s⁻¹. When the basket is 58.8 m above the ground, a sandbag is released from the basket. The sandbag then moves freely under gravity. The particle model is used, air resistance is ignored and g = 9.8 m s⁻².

A delivery van is travelling along a straight road at 18 m s⁻¹ when the driver sees a fallen branch lying across the road 45 m ahead. The van continues at 18 m s⁻¹ for a reaction time of 0.7 s, and then the driver brakes. While braking, the van decelerates uniformly at 4.5 m s⁻². The van is modelled as a particle.

An electric go-kart moves along a straight track. It starts from rest at time t = 0. Figure 1 shows its acceleration–time graph for 0 ≤ t ≤ 14, where t is in seconds. The go-kart is modelled as a particle.

A puck is projected up a straight ramp from a point A at the bottom of the ramp. It moves along a line of greatest slope with constant acceleration. Figure 1 shows its velocity–time graph for the first 6 s, taking the direction up the ramp as positive. The ramp is long enough for the puck not to reach the top. The puck is modelled as a particle.

A ball is thrown vertically upwards from a point on the top of a wall and later falls past the wall to the ground. Figure 1 shows the graph of its height h metres above the ground against the time t seconds after it is thrown, until it reaches the ground. The highest point (1, 19.6) and the point (3, 0) are marked on the graph. The particle model is used, air resistance is ignored and g = 9.8 m s⁻².

A ball is released from rest at time t = 0 and falls vertically onto a horizontal floor. It bounces and rises vertically, and then falls back to the floor. Figure 1 shows its velocity–time graph for 0 ≤ t ≤ 2.4, taking downwards as positive; the marked points have the coordinates shown. The time for which the ball is in contact with the floor is negligible. The particle model is used, air resistance is ignored and g = 9.8 m s⁻².

In a fitness test a runner starts from rest on a line A, runs in a straight line to a parallel line B, and then runs straight back to A, where she stops. She is instantaneously at rest at B when t = 6. Figure 1 shows her speed–time graph for the whole test, 0 ≤ t ≤ 13, where t is in seconds. The runner is modelled as a particle.

Figure 1 shows a diver on a springboard. She leaves the end of the board moving vertically upwards with speed 4.9 m s⁻¹ and moves freely under gravity until she enters the water. In the water she moves vertically downwards with constant deceleration and comes to rest at the depth shown. The height of the end of the board above the water surface is shown in the figure. The diver is modelled as a particle moving in a vertical line that does not meet the board; in the air, air resistance is ignored and g = 9.8 m s⁻².

A particle P moves along a straight line with constant acceleration. At time t = 0, P passes through a fixed point O on the line, moving with speed 9 m s⁻¹ in the positive direction. The acceleration of P is 2 m s⁻² in the negative direction. The displacement of P from O at time t seconds is s metres.

A motorcycle is at rest at a junction on a straight road. At time t = 0 a bus passes the junction, travelling along the road at a constant speed of 16 m s⁻¹. At t = 3 s the motorcycle sets off after the bus, moving with constant acceleration 2 m s⁻² until it reaches its top speed of 28 m s⁻¹. It then continues at 28 m s⁻¹. Both vehicles are modelled as particles moving along the same straight line.

Ball A is projected vertically upwards from a point X on the ground with speed 24.5 m s⁻¹. At the same instant, ball B is released from rest at a point H metres vertically above X. The balls move in the same vertical line. Both balls are modelled as particles, air resistance is ignored and g = 9.8 m s⁻². In parts (a) to (c), H = 49.

A test vehicle on a straight track starts from rest and accelerates uniformly at a m s⁻² until its speed is 30 m s⁻¹. It then travels at 30 m s⁻¹ for a time, and then decelerates uniformly at 2a m s⁻² until it comes to rest. The whole journey takes 100 s and covers a distance of 2.55 km. The vehicle is modelled as a particle.

A coach passes a junction J at time t = 0, travelling along a straight road and slowing down for roadworks ahead. At the same instant an ambulance starts from rest at J and moves along the same road in the same direction. Figure 1 shows the velocity–time graphs of the coach and the ambulance for 0 ≤ t ≤ 25, where t is in seconds. Both vehicles are modelled as particles.

Two boats, P and Q, move in the same direction along a straight canal. At time t = 0 they are in the positions shown in Figure 1, with the velocities shown. The engine of P has been switched off, so P decelerates uniformly at the rate shown until it comes to rest, and then remains at rest. Q moves at constant speed throughout. The boats are modelled as particles moving along parallel straight lines, and t is in seconds.

A train travels along a straight track from rest at station P to rest at station Q, a distance of 7.5 km. Figure 1 is a sketch of its speed–time graph, showing its speeds in m s⁻¹ and the duration in seconds of each of the five stages. The train travels at its greatest speed for T seconds, then slows to pass through a section of track with a speed limit, and each change of speed is at a constant rate. The train is modelled as a particle.

Figure 1 shows the displacement–time graph for a particle R, which moves on a straight line with constant acceleration and is at the fixed point O on the line at time t = 0. Here s metres is the displacement of R from O at time t seconds, for 0 ≤ t ≤ 11. The points (2, 16) and (6, 24) lie on the curve.