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# Throwing a Ball

When you throw a ball straight up, does it spend more time going up, more time coming down, or the same time going up and down – taking into account both gravity and air resistance?

#### Solution

More time coming down.

Intuitive explanation:

The force due to gravity is constant in both magnitude and direction. Intuitively, it makes sense that, if the only force acting on the ball were gravity, the ball would take the same amount of time going up as coming down.

Air resistance, on the other hand, is not constant. Setting aside the fact that it is proportional to the velocity, more importantly, the direction is always opposite the direction the ball is moving. This means the ball reaches its peak faster, and falls down slower. Therefore, the ball should spend more time coming down if we take into account both gravity and air resistance.

Physics explanation, in terms of energy:

Pick an arbitrary interim height on the path of the ball going up and coming down.

The potential energy of the ball must be the same on its way up as on its way down, because it is at the same height.

However, because of air resistance, the total energy (potential energy plus kinetic energy) must be less on its way down. Therefore, the speed (absolute value of the velocity) of the ball must be higher on the ball’s way up than on its way down.

Since this is true for all interim heights on the path of the ball, the ball is always a bit faster on its way up at every given height, and so it must spend more time on the way down.