You decide to launch a ball vertically so that a friend located 45 m above you can catch it. what is the minimum launch speed you can use? how long after the ball is launched will your friend catch it?

1) We can solve this part of the problem by using the law of conservation of energy. In fact, the initial energy of the ball is just kinetic energy:

where m is the mass of the ball and v is the initial speed of the ball. At its maximum height, the speed of the ball is zero, so its energy is just gravitational potential energy:

where g is the gravitational acceleration and h is the height above the ground.
Since the total energy must be conserved, we have

which means

If we use h=45 m, we find the minimum speed v such that the ball reaches an altitude of 45 m above the ground:


2) The motion of the ball is an accelerated motion, and the relationship between the distance covered by the ball and the time t is given by

where t is the time the ball takes to reach the altitude h=45 m. Rearranging the equation, we find