A basketball player throws his ball vertically upwards with an initial speed of v=40 m/s. Ignore air resistance. What is the speed of the ball at half of the maximum height?

Since we are neglecting air resistance the energy of the ball is conserved. We set the gravitational potential energy to be U=0 at h=0. Applying conservation of energy at h=0 and h=h_max , we get: U₁ + K₁=U₂ +K₂(1), at h=0 the potential energy is U₁=0 since we did set it so and at the maximum height the speed is 0, therefore K₂=0. So, (1) becomes mv²/2=mgh_max (2).

Now applying conservation of energy at h=0 and h=h_max/2: mu²/2+mgh_max/2=mv²/2, and using (2) we get, mu²/2 +mv²/4=mv²/2, which simplifies to: u²=v²/2, therefore the speed of the ball at h=h_max /2 is u=v/sqrt(2)=28.28 m/s