# From a lift moving upward with a uniform acceleration a, a man throws a ball vertically upwards...

## Question:

From a lift moving upward with a uniform acceleration a, a man throws a ball vertically upwards with a velocity v relative to the lift. The ball comes back to the man after a time t. Show that a + g = 2 v/t.

## Pseudo Force:

When the motion of an object is described using a non-inertial reference frame (a frame having non-zero acceleration) then, an apparent force acts on the object. This apparent force is known as Pseudo force.

It is also called as fictitious force, inertial force or d'Alembert force.

Taking upward direction as positive.

Given:

• The acceleration of the lift is {eq}a {/eq}.
• The ball comes back to the man after a time t.

The free-body diagram of the ball is shown in the figure below.

In the above figure,

• {eq}v {/eq} is the velocity of the ball relative to the lift.
• a is the acceleration of the lift.
• mg is the weight of the ball.
• ma is the pseudo force on the ball when observed by an observer sitting on the lift.

So, the acceleration of the ball relative to the lift is;

{eq}\begin{align} a_{ball} &= -\dfrac{ma+mg}{m}\\ &=-(a+g)\\ \end{align} {/eq}

Frome equation of motion along vertically upward with respect to the lift;

{eq}\begin{align} -v &= v+a_{ball} t\\ \implies -v &= v-(a+g) t\\ \implies a+g &= \dfrac{2v}{t}\\ \end{align} {/eq}

Hence proved.