(a) State the principle of conservation of linear momentum.
(b) Explain the mode of action of a propelled rocket.
(c) During a training session, two footballers pass a ball repeatedly between themselves. Give two reasons why the to and fro motion of the ball is not simple harmonic.
(d) A ball is dropped from a height, at the same time as another ball is projected horizontally from the same height.
(i) Would the balls hit the ground at the same time?
(ii) Explain your answer in (i).
(e) A ball of mass 0.10 kg is projected horizontally onto a vertical wall with a speed of 17 ms\(^{-1}\). The ball makes contact with the wall for 0.15 s and rebounds horizontally with a speed of 13 ms\(^{-1}\).
Calculate the:
(i) change in momentum of the ball;
(ii) average force exerted on the ball during its collision with the wall.
(a) Principle of conservation of linear momentum
In a closed (isolated) system on which no net external force acts, the total linear momentum of the bodies before an interaction (such as a collision) is equal to the total linear momentum after the interaction.
(b) Mode of action of a propelled rocket
A rocket burns fuel and expels a large mass of hot exhaust gases backwards at very high speed. By the conservation of momentum, the backward momentum given to the gases is balanced by an equal and opposite forward momentum gained by the rocket, so the rocket is driven forward. (This is also Newton's third law: action and reaction.)
(c) Two reasons the to-and-fro motion of the ball is not simple harmonic
- There is no restoring force that is proportional to, and directed opposite to, the displacement from a fixed equilibrium position.
- The motion is not periodic about a fixed point with a fixed period; the ball travels between the players at roughly constant speed rather than with sinusoidally varying speed.
(d)(i) Yes, both balls hit the ground at the same time.
(d)(ii) The vertical motion is independent of the horizontal motion. Both balls start with the same vertical velocity (zero) and fall under the same acceleration due to gravity g through the same height, so \( h = \tfrac{1}{2}g t^2 \) gives the same time of fall for each, regardless of the horizontal projection.
(e) Take the initial direction of motion as positive; the ball rebounds in the opposite direction.
(i) Change in momentum
\[ \Delta p = m(v - u) = 0.10 \times (-13 - 17) = 0.10 \times (-30) = -3.0\,\text{kg m s}^{-1} \]
The change in momentum is 3.0 kg m s\(^{-1}\) (directed away from the wall).
(ii) Average force
\[ F = \frac{\Delta p}{t} = \frac{3.0}{0.15} = 20\,\text{N} \]
The average force exerted on the ball is 20 N (directed away from the wall).
(a) Principle of conservation of linear momentum
In a closed (isolated) system on which no net external force acts, the total linear momentum of the bodies before an interaction (such as a collision) is equal to the total linear momentum after the interaction.
(b) Mode of action of a propelled rocket
A rocket burns fuel and expels a large mass of hot exhaust gases backwards at very high speed. By the conservation of momentum, the backward momentum given to the gases is balanced by an equal and opposite forward momentum gained by the rocket, so the rocket is driven forward. (This is also Newton's third law: action and reaction.)
(c) Two reasons the to-and-fro motion of the ball is not simple harmonic
- There is no restoring force that is proportional to, and directed opposite to, the displacement from a fixed equilibrium position.
- The motion is not periodic about a fixed point with a fixed period; the ball travels between the players at roughly constant speed rather than with sinusoidally varying speed.
(d)(i) Yes, both balls hit the ground at the same time.
(d)(ii) The vertical motion is independent of the horizontal motion. Both balls start with the same vertical velocity (zero) and fall under the same acceleration due to gravity g through the same height, so \( h = \tfrac{1}{2}g t^2 \) gives the same time of fall for each, regardless of the horizontal projection.
(e) Take the initial direction of motion as positive; the ball rebounds in the opposite direction.
(i) Change in momentum
\[ \Delta p = m(v - u) = 0.10 \times (-13 - 17) = 0.10 \times (-30) = -3.0\,\text{kg m s}^{-1} \]
The change in momentum is 3.0 kg m s\(^{-1}\) (directed away from the wall).
(ii) Average force
\[ F = \frac{\Delta p}{t} = \frac{3.0}{0.15} = 20\,\text{N} \]
The average force exerted on the ball is 20 N (directed away from the wall).
