Batman, whose mass is 90 kg, falls straight down from a bridge into a boat of mass 506 in which a criminal is fleeing. Initially, the speed of the boat is 10 m/s. What is the speed of the boat after Batman lands on it?
To find the speed of the boat after Batman lands on it, we can use the principle of conservation of momentum.
The principle of conservation of momentum states that the total momentum of an isolated system remains constant if no external forces act on it. In this case, the boat and Batman form an isolated system.
The equation for momentum is:
momentum = mass × velocity
Before Batman lands on the boat, the boat's momentum can be calculated as:
momentum of boat before = mass of boat × velocity of boat
Given that the mass of the boat is 506 kg and the initial velocity of the boat is 10 m/s, we can calculate the momentum of the boat before Batman lands:
momentum of boat before = 506 kg × 10 m/s
Now, let's consider Batman. Since Batman is falling straight down, his velocity just before landing on the boat can be calculated using the equation for gravitational potential energy:
potential energy = mass × gravitational acceleration × height
In this case, the gravitational acceleration is approximately 9.8 m/s², and Batman fell from a bridge, so we assume the height is reasonably high. Since Batman's mass is given as 90 kg, we can calculate his velocity just before landing using the potential energy equation:
potential energy = 90 kg × 9.8 m/s² × height
To simplify the calculation, we'll assume Batman's height is negligible compared to the bridge's height, so his potential energy just before landing is negligible compared to his kinetic energy. We can set potential energy to zero and solve for his velocity:
0 = 90 kg × 9.8 m/s² × height
The height cancels out, and we get:
0 = 882 kg·m²/s²
Since the potential energy is zero, Batman's kinetic energy just before landing is converted into the boat's kinetic energy after landing. Therefore, we can set the boat's final momentum equal to Batman's initial momentum and solve for the boat's final velocity:
momentum of boat after = mass of boat × velocity of boat after
momentum of boat after = 506 kg × velocity of boat after
momentum of boat after = momentum of Batman before
momentum of Batman before = mass of Batman × velocity of Batman before
momentum of Batman before = 90 kg × velocity of Batman before
Setting these equal, we have:
506 kg × velocity of boat after = 90 kg × velocity of Batman before
Solving for the velocity of the boat after Batman lands, we get:
velocity of boat after = (90 kg × velocity of Batman before) / 506 kg
Substituting the value for the velocity of Batman before (obtained from the potential energy equation) into this equation will give us the final answer for the speed of the boat after Batman lands on it.