# What happens to the momentum of earth-ball system when a ball is thrown upwards?

I would refer to linear momentum wherever I have used the word momentum.

I saw a question which was given in my textbook:-

When a ball is thrown upwards from the surface of earth,then:-

1. Earth and ball move towards each other.
2. Earth is stationary and ball moves upwards.
3. Earth and ball move away from each other.
4. None of these.

Answer:- (3) Earth and ball move away from each other to conserve momentum.

Following is the sequence of events which a I went through:-

Event 1

I read the question and tried to solve it.I thought that the answer would be none of these because first of all question is quite unclear because it does not describes that in which way the ball is thrown upwards?I thought of the following specific case:- In this image a man is holding a ball in his hand.He may throw it upwards.

If we take earth and ball as our system then the external forces on this system are

1. Normal reaction on earth due to man's weight.

2. The force with which he throws the ball up on ball.

With these thoughts I supposed that linear momentum of this system is not conserved.

At the very next moment I assumed that the man is massless(having negligible mass),then the normal reaction between earth and man vanishes but there is still a net external force which is the force exerted by man's hand on ball.

With analysis I concluded that momentum of system is not conserved.

Event 2

I asked my teacher about my assumptions.He said they are correct but he said that the momentum is still conserved.He told the reason that when the ball is thrown upwards then the contact time between hand and ball is very small as well as the contact force is too small such that the impulse imparted to the system is very small which can be neglected.

Event 3

I again analysed my teacher's arguments and concluded:-

1. Even if the impulse is small but momentum of the system still changes.Even if the ball is of $$0.5 kg$$ and is thrown upwards with a speed if $$6\frac{m}{s}$$ then the change in momentum is $$3\frac{kg.m}{s}$$.
2. If the ball is too heavy say 10 million kilograms and man is strong enough to throw it upward with a speed of $$10^{-5}\frac{m}{s}$$ then still momentum change is $$10\frac{kg.m}{s}$$.

With these arguments I am quite confused whether the linear momentum will be conserved or not?

• $$\frac{d\boldsymbol{P} } {dt}= \boldsymbol{F}$$,If $\boldsymbol{F}=0$ then angular momentum $\boldsymbol{P}$ be conserved. Nov 19 '19 at 15:06
• Nov 19 '19 at 15:42

The situation you are talking about can be visually represented by the following system with both the objects at the start possessing the same momentum. Hence law of conservation of momentum is not violated.