A 50.0 kg soccer player jumps vertically upwards and heads the 0.45 kg ball as it is descending vertically with a speed of 27.0 m/s. If the player was moving upward with a speed of 2.80 m/s just before impact. (a) What will be the speed of the ball immediately after the collision if the ball rebounds vertically upwards and the collision is elastic?
(Note that the force of gravity may be ignored during the brief collision time.)
To determine the speed of the ball immediately after the collision, we can use the principle of conservation of momentum. According to this principle, the total momentum before the collision must be equal to the total momentum after the collision, assuming no external forces are involved.
Step 1: Find the initial momentum of the soccer player and the ball separately.
The initial momentum of an object can be calculated by multiplying its mass by its velocity.
The initial momentum of the player (m1) is given by:
momentum1 = mass1 × velocity1
= 50.0 kg × 2.80 m/s
The initial momentum of the ball (m2) is given by:
momentum2 = mass2 × velocity2
= 0.45 kg × 27.0 m/s
Step 2: Find the final momentum of the soccer player and the ball separately.
Since the player is moving vertically upwards, the velocity of the player after the collision is in the opposite direction with a magnitude of 2.80 m/s.
The final momentum of the player (m1) is given by:
momentum1f = mass1 × velocity1f
= 50.0 kg × (-2.80 m/s)
Since the ball rebounds vertically upwards, its velocity after the collision is also in the opposite direction with a magnitude that we need to find.
Let's represent the final velocity of the ball as v2f.
The final momentum of the ball (m2) is given by:
momentum2f = mass2 × velocity2f
= 0.45 kg × v2f
Step 3: Apply the conservation of momentum principle.
According to the conservation of momentum, the total momentum before the collision must be equal to the total momentum after the collision.
Initial momentum (m1 + m2) = Final momentum (m1f + m2f)
Substituting the values we have calculated:
(50.0 kg × 2.80 m/s) + (0.45 kg × 27.0 m/s) = (50.0 kg × (-2.80 m/s)) + (0.45 kg × v2f)
Step 4: Solve for v2f.
Rearranging the equation and solving for v2f:
0.45 kg × v2f = (50.0 kg × 2.80 m/s) + (0.45 kg × 27.0 m/s) - (50.0 kg × (-2.80 m/s))
v2f = [(50.0 kg × 2.80 m/s) + (0.45 kg × 27.0 m/s) - (50.0 kg × (-2.80 m/s))] / 0.45 kg
Now, you can plug in the values and calculate v2f.