To solve this problem, we can use the law of conservation of momentum to determine the velocities of the pendulum bobs after the collision.
First, we need to find the initial momentum of the lighter bob before the collision:
Initial momentum = mass * velocity
= 75 g * 0 m/s (since the bob is released from rest)
= 0 kg * m/s
After the collision, the total momentum of the two bobs is still 0 kg * m/s, since momentum is conserved. Therefore, the final momentum of the heavier bob must be equal in magnitude but opposite in direction to the initial momentum of the lighter bob.
Final momentum of heavier bob = -initial momentum of lighter bob
= -(0 kg * m/s)
= 0 kg * m/s
Since the final momentum of the heavier bob is 0 kg * m/s, its velocity after the collision must also be 0 m/s.
The final velocity of the lighter bob can be found using the law of conservation of momentum:
Final momentum = initial momentum
= (mass of lighter bob * velocity of lighter bob) + (mass of heavier bob * velocity of heavier bob)
Substituting in the known values and solving for the velocity of the lighter bob, we find:
(75 g * velocity of lighter bob) + (95 g * 0 m/s) = 0 kg * m/s
velocity of lighter bob = -0.8 m/s
To find the height to which each bob rebounds, we can use the equation for the height of a pendulum:
Height = L * (1 - cos(θ))
where L is the length of the pendulum and θ is the angle of the pendulum from its initial position.
Since the lighter bob is released from rest at an angle of 90° (horizontal), its initial angle θ is 90°. Substituting in the known values, we find:
Height = 0.85 m * (1 - cos(90°))
= 0.85 m * (1 - (-1))
= 0.85 m * 2
= 1.7 m
The heavier bob does not gain any height, since it is stationary after the collision. Therefore, the lighter bob will rebound to a height of 1.7 m, while the heavier bob will remain at the bottom of its arc.