Projectile motion

A person kicks a stationary football horizontally.

During the kick, the person gives an impulse of 6.55 N s to the ball during a contact time of 12.0 ms. The mass of the football is 410 g. Assume that friction forces are negligible.

Calculate the kinetic energy transferred to the ball during the kick.

Figure 8 shows how the force F acting on the ball varies with time t.

Determine the scale for the F / N axis and add your scale to Figure 8.

A player jumps up to head a football when it is 2.0 m above the ground. The initial velocity of the ball is 18 m s⁻¹ at an angle of 48° to the horizontal.

Figure 9 shows the path of the ball above horizontal ground.

Determine the time that the ball takes to reach the ground from the instant it leaves the player’s head.

Assume that the frictional forces acting on the ball are negligible.

A skydiver is falling at a terminal velocity v₁. When she is at a height h₁ she opens her parachute. When she is at a height h₂ she reaches a new terminal velocity v₂.

Which is true?

A projectile is fired from ground level over horizontal ground.

Its initial velocity is u at an angle θ to the horizontal.

The range of the projectile is d.

A second projectile is fired with a velocity 2 u at the same angle.

What is the range of this projectile? Assume that air resistance is negligible.

A particle travelling horizontally at 1.0 × 10⁷ m s⁻¹ enters a region where it has a constant

What is the horizontal distance the particle has travelled in the region when its vertical

An object is thrown vertically upwards at time t = 0

The object reaches its maximum height when t = T and reaches its terminal speed on the way down.

The magnitude of the object’s acceleration is a.

Which graph shows the variation of a with t ?

A ball is kicked from point P on level ground. The ball initially travels at 45° to the horizontal. The ball reaches its maximum height after a time of 2.0 s. Air resistance can be ignored.

What is the displacement of the ball from P when at its maximum height?

A sphere of mass m falls with speed v. The resistive force on the sphere is kv, where k is a constant.

What is the terminal speed of the sphere?

A stroboscope emits bright flashes of white light. The duration of each flash and the frequency of the flashes can be varied.

Table 1 shows information about the stroboscope.

The duration of each flash is T₁. The time from the start of a flash to the start of the next flash is T₂.

The duty cycle of a stroboscope is defined as .

What is the maximum duty cycle of the stroboscope?

Tick () one box.

Figure 1 shows images produced in an experiment in which a bouncing ball is illuminated by a stroboscope. The stroboscope flashes at a constant frequency.

Suggest why T₁ must be very short for this experiment.

Figure 2 shows the first six images starting with n = 0, where n is the image number.

The images are used to determine: H, the vertical distance from the bottom of the ball to the floor when n = 0 h, the vertical distance from the bottom of the ball to the floor for each non-zero value of n.

The n = N image is produced at the instant that the ball hits the floor for the first time. For n between 0 and N it can be shown that

where u₀ is the vertical velocity of the ball when n = 0 g is the acceleration due to gravity f is the frequency of the flashes.

In order to find g, a graph is plotted with values of on the y

Suggest what is plotted on the x-axis. Go on to explain how g is determined from this graph.

The following data are recorded.

H = 1550 mm f = 31.0 Hz

The graphical analysis of data from Figure 1 gives g as 9.79 m s⁻².

Determine u₀.

Figure 3 shows positions of the bottom of the ball for n = 40 to n = 66 In this range of positions, the ball makes contact with the floor for the second and third times. Values of h, the vertical distance from the bottom of the ball to the floor, are plotted on the y-axis. Values of s, the horizontal displacement from a point on the floor below the centre of the n = 0 image, are plotted on the x-axis.

Determine, in mm s⁻¹, the horizontal velocity of the ball between the second and third contacts of the ball with the floor.

Determine the time between the second and third contacts. Annotate Figure 3 to show your method.

Figure 8 shows the H-shaped posts used in a game of rugby.

Figure 9 shows the path of a ball that is kicked and just passes over the crossbar. The initial velocity of the ball is 20.0 m s⁻¹ at an angle of 40.0° to the ground.

You should consider air resistance to be negligible and treat the ball as a simple projectile.

The top of the crossbar is 3.00 m above the horizontal ground.

Show that the minimum speed of the ball in flight is about 15 m s⁻¹.

Explain your answer.

The ball just passes over the crossbar at a time t after it is kicked.

Show that t must satisfy the following equation:

There are two solutions to the equation

Discuss which of the two solutions is the time taken for the ball to pass over the crossbar from when it is kicked.

In your answer you should

• state the value for t given by each solution

• explain the physical significance of the other solution.

Another attempt is made to kick the ball over the crossbar. The initial velocity of the ball is the same as in the first attempt. This kick is made from a horizontal distance of 38 m from the posts.

Deduce whether the ball can pass over the crossbar.

Figure 10 shows the variations with time of the vertical velocity of a ball with and without air resistance.

Discuss the features of the motion of the ball shown by the two graphs.

In your answer you should refer to

• the gradients of the graphs

• the area between each line and the time axis.

A coin is projected horizontally from the top of a desk. The diagram shows the coin at one point in its path. The air resistance is negligible.

The arrows E, F and G represent different directions.

Which row gives the direction of the acceleration and the direction of the momentum of the coin at this point?

Figure 5 shows a simplified catapult used to hurl projectiles a long way.

The counterweight is a wooden box full of stones attached to one end of the beam. The projectile, usually a large rock, is in a sling hanging vertically from the other end of the beam. The weight of the sling is negligible. The beam is held horizontal by a rope attached to the frame.

The catapult is designed so that the weight of the beam and the weight of the empty wooden box have no effect on the tension in the rope.

Suggest how the pivot position achieves this.

The stones in the counterweight have a total mass of 610 kg and the projectile weighs 250 N.

Calculate the tension in the rope.

When the rope is cut, the counterweight rotates clockwise. When the beam is vertical it is prevented from rotating further. The projectile is then released horizontally with a velocity of 18 m s^–1, as shown in Figure 6.

The projectile is released at a height of 7.5 m above ground level.

The range of the catapult is the horizontal distance between the point where the projectile is released to the point where it lands.

Calculate the range. Ignore air resistance.

In another release, the sling is adjusted so that a projectile of the same mass is released just before the wooden beam is vertical. The projectile is not released horizontally.

Discuss the effect this change has on the range of the catapult.

A projectile is launched with a speed of 25 m s⁻¹ at an angle of 35° to the horizontal, as shown in the diagram.

Air resistance is negligible.

What is the time taken for the projectile to return to the ground?

What is the horizontal distance from the starting point of the projectile when it hits the ground?