Projectiles and variable acceleration – Year 13 A level maths worksheet with answers to download as PDF

A ball is projected from level ground with speed 24 m s\(^{-1}\) at \(40^\circ\) above the horizontal. Take \(g=9.8\) m s\(^{-2}\).

1. Find the initial horizontal and vertical components of the velocity. [2 marks]
2. Find the time taken to reach the highest point. [2 marks]
3. Find the greatest height reached. [2 marks]

A ball rolls off the edge of a horizontal table of height 1.2 m with speed 3 m s\(^{-1}\).

1. Find the time taken to reach the floor. [2 marks]
2. Find the horizontal distance from the table edge to the landing point. [1 mark]
3. Find the speed and direction of the ball as it lands. [4 marks]

A golf ball is hit from level ground with speed 30 m s\(^{-1}\) at \(25^\circ\) above the horizontal.

1. Find the time of flight. [2 marks]
2. Find the range. [2 marks]
3. Find the greatest height. [2 marks]

A particle moves in a straight line. At time \(t=0\) it is at the origin with velocity 3 m s\(^{-1}\). Its acceleration at time \(t\) seconds is \(a=4t-6\) m s\(^{-2}\).

1. Find an expression for the velocity. [3 marks]
2. Find an expression for the displacement from the origin. [3 marks]
3. Find the velocity and displacement when \(t=2\). [2 marks]

A particle moves in a straight line with velocity \(v=3t^2-12t+9\) m s\(^{-1}\) for \(t\ge0\).

1. Find the acceleration at time \(t\). [2 marks]
2. Find the times at which the particle is instantaneously at rest. [2 marks]
3. Find the acceleration at each of those times and describe the motion. [3 marks]

A particle moves in a straight line with velocity \(v=t^2-6t+8\) m s\(^{-1}\) for \(0\le t\le4\).

1. Find the times when \(v=0\). [2 marks]
2. Find the displacement of the particle from \(t=0\) to \(t=4\). [3 marks]
3. Find the total distance travelled in the 4 seconds. [3 marks]

A footballer takes a free kick from level ground 20 m from the goal line. The ball leaves the ground at 18 m s\(^{-1}\) at \(22^\circ\) above the horizontal. A defensive wall 2 m high stands 9.15 m from the ball. The crossbar is 2.44 m high. Model the ball as a particle.

1. Find the height of the ball when it is above the wall, and decide if it clears the wall. [4 marks]
2. Find the height of the ball when it reaches the goal line. [3 marks]
3. Does the ball go under the crossbar? [1 mark]
4. State two factors that the model ignores and say how each would affect the answer. [2 marks]

A diver runs off a horizontal cliff top of height 25 m with horizontal speed 2 m s\(^{-1}\). Rocks extend 3 m out from the base of the cliff.

1. Find the time taken to reach the water. [2 marks]
2. Show that the diver clears the rocks. [2 marks]
3. Find the speed and angle of entry into the water. [4 marks]

Water leaves a nozzle at ground level with speed 8 m s\(^{-1}\) at \(60^\circ\) to the horizontal. Treat the water as a particle.

1. Find the maximum height of the jet. [3 marks]
2. Find the horizontal range. [3 marks]
3. The designer wants to double the range at the same angle. By what factor must the speed increase? [2 marks]

A cricket ball is hit from a height of 1 m above the ground with speed 15 m s\(^{-1}\) at \(35^\circ\) above the horizontal.

1. Find the greatest height of the ball above the ground. [3 marks]
2. Find the time at which the ball reaches the ground. [4 marks]
3. Find the horizontal distance from the batter to the point where the ball would land. [2 marks]

The velocity of a lift in a tall building is modelled by \(v=0.5t-0.05t^2\) m s\(^{-1}\) for \(0\le t\le10\) seconds, where \(t\) is time since the lift left the ground floor.

1. Show that the lift is at rest at \(t=0\) and \(t=10\). [2 marks]
2. Find the maximum speed. [3 marks]
3. Find the height gained during the 10 seconds. [3 marks]
4. Find the acceleration at \(t=8\) and explain its sign. [2 marks]

A jogger runs along a straight track. Her velocity at time \(t\) minutes is \(v=t^2-8t+12\) m s\(^{-1}\) for \(0\le t\le6\), where positive is towards the park gate.

1. Find the times at which she turns round. [2 marks]
2. Show that her displacement from the start after 6 minutes is zero. [3 marks]
3. Find the total distance run. [3 marks]

A train accelerates from rest at the station. Its acceleration is modelled by \(a=2-0.1t\) m s\(^{-2}\), for \(0\le t\le20\), where \(t\) is in seconds.

1. Find an expression for the velocity of the train. [3 marks]
2. Find the speed when the acceleration becomes zero. [2 marks]
3. Find the distance covered in the first 20 s. [3 marks]

A drone moves so that its position vector relative to a fixed point is \(\mathbf r=2t\mathbf i+(t^2-4t)\mathbf j\) metres at time \(t\) seconds.

1. Find the velocity and acceleration at time \(t\). [3 marks]
2. Find the speed at \(t=3\). [2 marks]
3. Find the time when the speed is least and the least speed. [3 marks]

In a basketball game a player releases the ball at a height of 2.1 m, at \(50^\circ\) above the horizontal. The hoop is at a height of 3.05 m and the centre of the hoop is 4.6 m horizontally from the release point. Model the ball as a particle.

1. Show that the path of the ball can be written \(y=x\tan\theta-\dfrac{gx^2}{2u^2\cos^2\theta}\), where \(y\) is the height above the release point. [4 marks]
2. Find the speed of release that takes the ball through the centre of the hoop. [5 marks]
3. Find the time taken for the ball to reach the hoop. [2 marks]
4. The ball is released 0.1 m higher but with the same speed. Describe qualitatively how the ball reaches the hoop. [1 mark]

A sports scientist fires balls from level ground at 25 m s\(^{-1}\) at several angles. The theoretical ranges (no air resistance) are in the table.

1. Verify the range for \(\theta=30^\circ\). [2 marks]
2. Explain, using \(R=\dfrac{u^2\sin2\theta}g\), why the ranges for 15° and 75° are the same and why 45° is best. [3 marks]
3. Find the two possible angles for a range of 50 m. [4 marks]
4. A real ball launched at 30° travels 52 m. Compare it with the model and give a reason. [2 marks]

A particle moves along a straight line. Its velocity is \(v=t^3-6t^2+9t\) m s\(^{-1}\) at time \(t\) seconds, for \(t\ge0\). At \(t=0\) it is at the origin \(O\).

1. Show that \(v=t(t-3)^2\) and describe when the particle is at rest. [3 marks]
2. Find the displacement at time \(t\). [3 marks]
3. Find the greatest speed for \(0\le t\le3\). [4 marks]
4. Find the acceleration and displacement when \(t=4\). [3 marks]

A stone is thrown from the top of a cliff 30 m above sea level with speed 18 m s\(^{-1}\) at \(40^\circ\) above the horizontal.

1. Find the greatest height above the sea. [3 marks]
2. Find the time taken to hit the sea. [4 marks]
3. Find the horizontal distance from the foot of the cliff to the landing point. [2 marks]
4. Find the speed and the direction of motion when the stone hits the sea. [4 marks]

The path of a jet of water from a sprinkler is observed to be \(y=x-0.05x^2\) for \(y\ge0\), where \(x\) and \(y\) are measured in metres from the nozzle.

1. Find the angle of projection. [2 marks]
2. Use the equation \(y=x\tan\theta-\dfrac{gx^2}{2u^2\cos^2\theta}\) to find the speed of projection. [4 marks]
3. Find the range and the greatest height of the jet. [4 marks]
4. The gardener wants to wet a flower 15 m away at a height of 1 m. Will the jet pass through that point? [2 marks]

A particle is projected from level ground with speed \(u\) at angle \(\theta\) above the horizontal. The range is \(R\) and the greatest height is \(H\).

1. Show that \(H=\dfrac{R\tan\theta}4\). [4 marks]
2. Check the result when \(u=20\) and \(\theta=30^\circ\). [2 marks]

A particle moves on a smooth horizontal surface in a liquid so that its acceleration is \(a=-0.5v\) m s\(^{-2}\). At \(t=0\) its velocity is 20 m s\(^{-1}\).

1. By writing \(a=\dfrac{dv}{dt}\) and separating variables, show that \(v=20e^{-0.5t}\). [4 marks]
2. Find the time taken for the speed to halve. [2 marks]
3. By integrating, show that the particle never travels more than 40 m. [4 marks]