Batman (mass = 87.3 kg) jumps straight down from a bridge into a boat (mass = 680 kg) in which a criminal is fleeing. The velocity of the boat is initially +14.8 m/s. What is the velocity of the boat after Batman lands in it?
To solve this problem, we can use the law of conservation of momentum, which states that the total momentum before an event is equal to the total momentum after the event.
The formula to calculate momentum is given by:
momentum = mass * velocity
In this case, we have two objects: Batman and the boat. Let's assume that the positive direction is upward.
Before Batman jumps, the boat has momentum given by:
momentum(boat before) = mass(boat) * velocity(boat)
= 680 kg * (-14.8 m/s) (since the boat is moving in the opposite direction of the positive direction)
Batman is initially at rest, so his momentum is:
momentum(Batman before) = mass(Batman) * velocity(Batman)
= 87.3 kg * 0
The total momentum before the event is equal to the sum of the momenta of the boat and Batman before:
total momentum before = momentum(boat before) + momentum(Batman before)
Now, Batman jumps into the boat. The boat and Batman now move together, so their combined momentum afterwards will be equal to the total momentum before.
After Batman jumps, the velocity of the boat and Batman together is denoted by Vf.
total momentum after = momentum(boat after) + momentum(Batman after)
= mass(boat + Batman) * Vf (since they will have the same final velocity)
From the conservation of momentum, we have:
total momentum before = total momentum after
Substituting the values we know:
momentum(boat before) + momentum(Batman before) = mass(boat + Batman) * Vf
(680 kg * (-14.8 m/s)) + (87.3 kg * 0) = (680 kg + 87.3 kg) * Vf
Simplifying the equation:
-10064 kg·m/s = 767.3 kg * Vf
Dividing both sides by 767.3 kg:
Vf = -10064 kg·m/s / 767.3 kg
≈ -13.1 m/s
Therefore, the velocity of the boat after Batman lands in it is approximately -13.1 m/s.
To find the velocity of the boat after Batman lands in it, we can apply the principle of conservation of momentum.
The principle of conservation of momentum states that the total momentum before an event is equal to the total momentum after the event, provided no external forces are acting on the system.
In this case, we can consider the system as Batman and the boat. The initial momentum of the system is given by the sum of the momenta of Batman and the boat before Batman jumps:
Initial momentum = (mass of Batman * velocity of Batman) + (mass of the boat * velocity of the boat)
Let's calculate the initial momentum of the system:
Initial momentum = (87.3 kg * 0 m/s) + (680 kg * 14.8 m/s)
Initial momentum = 0 + 10064 kg·m/s
Initial momentum = 10064 kg·m/s
According to the conservation of momentum, the total momentum after Batman lands in the boat should be equal to the initial momentum. Let's denote the final velocity of the boat as Vf.
Final momentum = (mass of Batman * velocity of Batman) + (mass of the boat * final velocity of the boat)
Since Batman is initially at rest (velocity = 0):
Final momentum = (87.3 kg * 0 m/s) + (680 kg * Vf)
Final momentum = 0 + 680 kg·m/s * Vf
Final momentum = 680 kg·m/s * Vf
According to the conservation of momentum:
Initial momentum = Final momentum
10064 kg·m/s = 680 kg·m/s * Vf
Solving for Vf:
Vf = 10064 kg·m/s / 680 kg
Vf ≈ 14.8 m/s
Therefore, the velocity of the boat after Batman lands in it is approximately 14.8 m/s