Acceleration Of A Ball thrown into the air
Tagged: acceleration, gravity, newtons, velocity

Acceleration Of A Ball thrown into the air

A ball sitting in a person’s hand is at rest. The ball is thrown into the air. There must be some upward acceleration that is greater than the acceleration due to gravity since in order for the ball to move, the upward force must be greater than the force of gravity. Since the mass of the ball doesn’t change, the acceleration upwards must be greater than the acceleration downwards for the upward force to be greater than the downward force.
The instant the ball leaves the hand, what is its acceleration?
A. Acceleration of gravity, downward. Upward acceleration from the throw. Something else
If a, how it can be that the acceleration changes instantaneously to that of gravity, what happens to the acceleration from the throw?
If b, at some point the only acceleration on the ball must be from gravity (because otherwise, the ball would keep going forever like a rocket), yet this means the acceleration is decreasing and at some point, it will be 0. Then there is no force acting on the ball and it is moving at a constant velocity, but this doesn’t make sense because the ball is certainly accelerating due to gravity.
If c, then what is really happening?
EDIT: It seems that according to the answer to another question here, “a” is correct. However, that question and its answer do not address what happens to the upward acceleration from the hand. Does it instantaneously become zero, such that the ball only has an initial velocity upward but not an initial acceleration upward?

First, we need to clear up the concept of acceleration. There is only one linear acceleration of the object. There is not an acceleration due to gravity and a separate acceleration due to the push of the hand. There are two forces (push of the hand and weight) that combine to give the actual, single acceleration. That’s Newton’s 2nd Law.
a = ΣF m.
a→=ΣF→m.
If the object accelerates upward, then the force from the hand upward on the object is greater than the weight downward. After the object completely loses contact with the hand, the only force is the weight, so the acceleration is downward at a magnitude equal to the local gravitational field due to the Earth:
a=1mGmMEarthR2Earth=GMEarthR2Earth=glocal.a=1mGmMEarthREarth2=GMEarthREarth2=glocal.
As the ball is leaving the hand, the hand exerts an ever decreasing force during a short period of time, but the total force, in reality, has a rapid, yet continuous change from net upward, through zero, to net downward and finally weight, mg(local) downward. At that point, the downward acceleration continues to cause the object to slow down, stop, and then fall.

Answer a is correct. As soon as the ball leaves the thrower’s hand, the only force acting on it is gravity, so its acceleration is gg downwards.
Note that while it is in the air the acceleration of the ball remains the same even though the velocity changes. It is −g−g when it leaves the hand, −g−g when it stops instantaneously at the top of its flight, and −g−g when it hits the ground. That is because the force on the ball mgmg remains constant (neglecting air resistance and the variation in gravity with distance from the centre of the Earth).
Acceleration can change instantaneously if a force is removed or applied instantaneously. In reality, the force from the thrower’s hand is removed gradually, reducing to zero as the ball leaves contact with the hand. This means that the ball is already decelerating before it leaves the thrower’s hand.
Perhaps you are thinking of velocity, which does cannot change instantaneously even if the acceleration does. An instantaneous change in velocity implies an infinite force, which cannot happen.
When physics is being taught the assumption is often made that forces are constant and removed instantaneously, in order to make calculations much easier. In reality, forces always change continuously and smoothly, if you look on a small enough time scale. For some purposes, the change might be so fast that, to a good approximation, it is effectively instantaneous.
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