Equations of Motion (CIE AS Physics)

Define acceleration.

A ball is kicked from horizontal ground towards a vertical wall, as shown in Fig. 1.1.

Fig. 1.1 (not to scale)

The horizontal distance between the initial position of the ball and the base of the wall is 24 m. The ball is kicked with an initial velocity v at an angle of 28° to the horizontal. The ball hits the top of the wall after a time of 1.5 s. Air resistance is negligible.

v_x = ................................................. m s^–1 [1]

[2]

time = ....................................................... s [2]

Fig. 1.2

[2]

maximum height = ...................................................... m  [2]

Calculate the mass of the ball.

mass = ..................................................... kg  [2]

A ball of greater mass is kicked with the same velocity as the ball in (b).

Air resistance is still negligible. 

State and explain the effect, if any, of the increased mass on the time taken by the ball to reach its maximum height.

A boy standing on a cliff shoots a pellet from his pellet gun vertically upwards. The variation of displacement s of the pellet with time t is shown in Fig. 1.1.

[2]

On Fig. 1.2, sketch the variation of velocity v of the pellet with time t.

Include your values from (a) on the axes.

Using Fig. 1.2

An identical pellet gun is fired horizontally off the edge of the cliff at the exact same time the other gun is fired vertically upwards. The vertically fired pellet is in motion for exactly 6.0 s before hitting the ground below the edge of the cliff.

Both pellets have the same initial speed.

A ball is projected horizontally at 27 m s^–1 from a vertical cliff. It travels a horizontal distance of 40 m before hitting the ground.

Assume that air resistance is negligible.

Calculate the vertical velocity of the ball just before it hits the ground.

Calculate the height of the cliff.

Sketch the graphs to show how the horizontal and vertical components of the velocity of the ball,  and  change with time t just before the ball hits the ground.

Label any appropriate values on the axes.

Calculate the resultant velocity of the ball just before it hits the ground.

A particle moves in a straight line with a constant acceleration, a. The initial velocity of the particle at t = 0 is u.  After a further time t, its velocity has increased to v.

(i)

Using the definition of acceleration, show that 

 

[2]

(ii)

Using the definition of average velocity, show that 

 

[3]

(iii)

Using these two equations, show that 

 

[3]

A student investigates the motion of a light ball bearing as it falls from rest alongside a vertical 100 cm scale.

The student sets up a camera to take flash photographs of the ball at 0.05 s intervals, as shown in Fig. 1.1.

The first photograph is taken at time t = 0.

Using Fig. 1.1, describe and explain the motion of the ball

The 100 cm scale is suspended 160 cm above the ground.

Determine the total time that the ball spends in free fall after it is dropped from the starting point of the 100 cm scale.

The student repeats the experiment with a lead sphere that falls with constant acceleration and does not reach a constant speed.

Determine the number of photographs, including the photograph at t = 0, that will be observed against the 100 cm scale.

An archer fires an arrow towards a target as shown in Fig. 1.1.

The centre of the target is at the same height as the initial position of the arrow.

The target is a distance of 100 m from the arrow. The arrow has an initial velocity of 62 m s^–1 and is fired at an angle of 11° to the horizontal.

Describe how the vertical and horizontal components of the arrow's velocity change from when it is fired until it reaches its maximum height. 

Assume that the effect of air resistance is negligible.

Show that the time taken for the arrow to reach its maximum height is about 1.2 s.

Calculate the horizontal distance, measured along the base line, by which the arrow misses the target.

During a fireworks display, a firework is launched into the air and explodes once it reaches a maximum height of 250 m.

The vertical component of the velocity at launch depends on both the initial velocity of the firework and the angle θ between the initial velocity and the horizontal.

The maximum initial velocity a firework can be launched at is 110 m s⁻¹.

On Fig. 1.1, sketch the variation of angle θ with the initial velocity required for the firework to reach the maximum height of 250 m. 

Include values from (a) on the axes.

Fig. 1.1

The length of the fuse within the firework is chosen to ensure it explodes just as it reaches its highest point above the ground.

The firework is launched into the air at an angle of 75° to the horizontal, as shown in Fig. 1.2.

Fig. 1.2

The firework contains a fuse with a burn rate of 1.5 cm s^–1.

Determine the length of the fuse used in the firework.

During the display, one of the fireworks is lit but falls over before launching. When it launches, it leaves the ground with an initial velocity of 75 m s⁻¹ at an angle of 30° to the horizontal, as shown in Fig. 1.3.

The firework moves in the direction of a hill. The side of the hill is inclined at 8.5° to the horizontal, as shown.

Fig. 1.3

Deduce whether the firework ignites before it lands on the hillside.

Fig. 1.1 shows a stunt motorcyclist descending down a ramp.

Fig. 1.1

The motorcyclist starts from rest at the top of the ramp at A and leaves the ramp at B horizontally.

After leaving the ramp at B, the motorcyclist lands on the ground at C, which is 6.4 m away. At C, the motorcyclist lands with a resultant velocity of 11.6 m s^–1 at an angle of 35° with the horizontal.

Calculate the initial velocity of the motorcyclist at B. 

In this question, assume that air resistance is negligible.

The motorcyclist practices a stunt on a different ramp inclined at 25º and 3 m high as shown in Fig. 1.2.

The other ramp is 24 m away and has a height of 1 m, as shown.

Fig. 1.2

Calculate the minimum initial velocity the motorcyclist requires in order to reach the other ramp.

The stunt motorcyclist's ultimate goal is to jump across a river and land on the other side. Fig. 1.3 shows the motorcyclist driving off a ramp at the edge of a river.

Fig. 1.3

The ramp is at an angle of 25° to the horizontal and the height at the end of the ramp is 3.0 m. The width of the river is 100 m. The initial velocity of the motorcyclist is 35 m s⁻¹.

Deduce whether the motorcyclist lands on the other side of the river.

Explain how air resistance would affect the jump.

An object is released near the surface of the Moon at time t = 0. Fig. 1.1 shows the variation of displacement s with time t of the object from the point of release. 

Fig. 1.1

Use the graph in Fig. 1.1 to determine a value for the acceleration of free fall close to the surface of the Moon.

Use Fig. 1.1 to estimate the instantaneous velocity of the object when t = 1.5 s.