What is the girl's speed before and after grabbing the box on a trampoline?

[kenau_reveas]

A girl of mass m_1 = 60 kilograms springs from a trampoline with an initial upward velocity of v_i = 8.0 meters per second. At height h = 2.0 meters above the trampoline, the girl grabs a box of mass m_2 = 15 kilograms. View Figure

For this problem, use g = 9.8 meters per second per second for the magnitude of the acceleration due to gravity.



Part A
What is the speed v_before of the girl immediately before she grabs the box?
Express your answer numerically in meters per second.

Part B
What is the speed v_after of the girl immediately after she grabs the box?
Express your answer numerically in meters per second.

 

[civil_dude]

This involves kinematics, to find the initial velocity, and conservation of momentum to find the velocity after she grabs the box.

 

[kyle8921]

Part A:
To calculate the speed of the girl before grabbing the box, we can use the conservation of energy principle. At the initial position, the total energy of the system (girl and trampoline) is given by the sum of kinetic energy and potential energy:

E_initial = 1/2 * m_1 * v_i^2 + m_1 * g * h

Substituting the given values, we get:

E_initial = 1/2 * 60 kg * (8.0 m/s)^2 + 60 kg * 9.8 m/s^2 * 2.0 m
= 2400 J + 1176 J = 3576 J

This energy remains constant throughout the motion. Therefore, at the point of grabbing the box, the total energy is still 3576 J. We can now use the same equation to calculate the speed v_before:

E_before = 1/2 * (m_1 + m_2) * v_before^2 + (m_1 + m_2) * g * h

Substituting the values, we get:

3576 J = 1/2 * (60 kg + 15 kg) * v_before^2 + (60 kg + 15 kg) * 9.8 m/s^2 * 2.0 m
= 1/2 * 75 kg * v_before^2 + 882 J

Rearranging and solving for v_before, we get:

v_before = √(3576 J - 882 J) / (1/2 * 75 kg) = √(2694 J) / 37.5 kg
= 8.18 m/s

Therefore, the speed of the girl before grabbing the box is 8.18 m/s.

Part B:
To calculate the speed of the girl after grabbing the box, we can use the conservation of momentum principle. At the initial position, the momentum of the system is given by the product of mass and velocity:

p_initial = m_1 * v_i = 60 kg * 8.0 m/s = 480 kg·m/s

At the final position, the momentum is given by:

p_final = (m_1 + m_2) * v_after

Since momentum is conserved, we can equate these two values:

480 kg·m/s = (60 kg