A golfer badly misjudges a putt, sending the ball only one-quarter of the distance to the hole. The original putt gave the ball an initial speed of V0. If the force of resistance due to the grass is constant, what would an initial speed in terms of V0 be needed to get the ball to the hole from its original position?

v = 2 v₀

Explanation:

For this exercise we will use the relationship between work and the change in kinetic energy

          W = ΔK

Where the work is of the friction force, which is always opposed to the movement whereby the angle is 180º

          W = - fr x

The distance for the first case is

         x = 1/4 L

We substitute

       -fr ¼ L = ½ m v₀²

        fr L = 2 m v₀²

In the second case, the new speed takes the ball the entire distance

        x = d

        -fr L = ½ m v²

We equal the two equations

        2 m v₀² = ½ m v²

        v² = 4 v₀²

        v = 2 v₀

that would be because of both gravity and friction. so     c