potential energy, momentum

Question

A basketball player is running at 4.60 m/s directly toward the basket when he jumps into the air to dunk the ball. He maintains his horizontal velocity.

What vertical velocity (in m/s) does he need to rise 0.600 meters above the floor?

How far (in m) from the basket (measured in the horizontal direction) must he start his jump to reach his maximum height at the same time as he reaches the basket?

Answer 1

Let

v = 4.60 m/s be the player's (constant) horizontal velocity,

w (to be computed) be his vertical velocity at the moment of jump,

h = 0.600 m be the maximum height by which his center of gravity rises,

t (unknown) be the time it takes to reach the height h,

m (unknown) be the player's mass,

s (to be calculated) be the distance from the basket where he needs to jump,

g = 9.81 m/² be gravitational acceleration.

We calculate w by conservation of energy.

His total mechanical energy at the point of the jump must equal his total mechanical energy

at the highest point.

For convenience, we can pick the floor level as the reference level for potential gravitation energy.

So before the jump his potential energy is 0, and

at the highest point his potential energy is mgh.

His kinetic energy depends on his speed.

At the highest point the vertical component of his velocity is 0, and

horizontal component is v, so

at the highest point the kinetic energy is mv²/2.

At the time of the jump his velocity is the vector sum of his horizontal velocity v and his vertical velocity w.

By Pythagorean Theorem the vector sum has magnitude √(v²+w²).

Therefore his kinetic energy at the moment of the jump is m(v²+w²)/2.

The expression of conservation of energy is the following equality between

the total mechanical energy at the point of jump and at the highest point:

m(v²+w²)/2 = mgh + mv²/2

From that

w = √(2gh)

Substituting actual numbers

w = √(2×9.81×0.6) = 3.43 m/s

To calculate s, we first need to calculate the time t

it takes to reach the maximum height.

That is the time it take gravity to reduce the velocity w to 0:

t = w/g

During this time t, his horizontal velocity v will carry him distance

s = vt.

So,

s = vw/g = v√(2gh)/g = v√(2h/g)

Substituting actual numbers

s = 4.6×√(2×0.6/9.81) = 1.6 m