Question

A ball is thrown vertically up with a velocity u . Another ball is thrown up with the same velocity but n sec later when and where do they meet

Answer 1

Answer:

Explanation:

Usually when we throw objects upwards or downwards on earth, we are the observers on earth. The motion of objects is w.r.t. 'us' who are standing in Earth's gravity in a non-inertial frame of reference. Hence we consider gravity force to be included in calculation in the form of 'g' in kinematic equations. (Ex: s=ut−12gt2)

But since this question demands motion of object A w.r.t object B and not w.r.t. 'us', we need not even consider the scenario in Earth's gravity, rather perform this experiment in space inside a non-inertial frame in space (just like Earth's gravity). In this frame all experiments must replicate the ones performed on earth as per Einstein's equivalence principle. So let's begin with our experiment in space.

Imagine yourself standing in a space vehicle. The vehicle is acclerating with a=9.8m/s2 in the direction of your head - i.e. upwards for you. You can feel your weight on legs due to floor pushing you upwards as the vehicle accelerates upwards - just how you feel your weight on earth due to gravity. In fact, Einstein claimed every experiment(even those involving clocks/time) will imitate as if being performed on Earth. This is because the space-time in earth's gravity is identical to space-time in our accelerating space ship(for observer's inside the ship). This is what Einstein's principle is.

So trusting Einstein, let's perform our experiment with objects A and B in our space vehicle. Also, I am floating in the space outside your vehicle and observing it - At some instance, when I see your vehicle having velocity v upwards as per my position, you throw ball A upwards with velocity u. So we both note different velocities for the ball -

Your observation : ball A initial speed = u

My observation : ball A speed = u+v

For me, your vehicle speed gets added to ball.

At that very same moment, your friend hanging from the ceiling of your vehicle leaves ball B. Since the vehicle is accelerating upwards with a=9.8m/s2,you will see ball B coming down with the same acceleration (just how you will see ball B falling on earth). But how do I see ball B? I see that when your friend left the ball B, he was moving with the vehicle at speed v, hence as per Newton's first law, ball B will continue to move at v. So I see ball B moving at speed v. So we both have our observations for ball B-

Your observation -

B is acclerating downward at a=9.8m/s2

My observation -

B is moving uniformly upward with speed v

You being in non-inertial reference, observe both balls accelerating, while me being an inertial reference observe both balls moving with uniform motion.

So my observation for both balls -

Ball A - Uniform motion = u+v upward

Ball B - Uniform speed = v upward.

Now answer to your question - what is speed of A relative to B?

Relative speed of A w.r.t B = (speed of A w.r.t me) - (speed of B w.r.t me)

=(u+v)−v

=u

Thus, ball B is seeing the ball A approaching it with constant speed u

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