Jennifer (mass 58.0 kg) is standing at the left end of a 13.0 m long 594.0 kg cart that has frictionless wheels and rolls on a frictionless track. Initially both Jennifer and the cart are at rest. Suddenly, Jennifer starts running along the cart at a speed of 5.70 m/s relative to the cart. How far will Jennifer have run relative to the ground when she reaches the right end of the cart?
In this situation, the center of mass of Jennifer and the cart (considered together) will remain in the same place.
The speed that she runs won't affect her position relative the the ground. It will only affect how soon she gets there.
Those hints should guide you easily to the solution.
To solve this problem, we need to use the concept of conservation of momentum. The initial momentum of the system (Jennifer + cart) is zero because both are at rest. However, when Jennifer starts running on the cart, she imparts momentum to the system.
Let's denote the direction Jennifer is running as positive. The momentum she imparts to the system is given by:
Momentum = mass * velocity
Jennifer's momentum relative to the ground is:
Initial momentum = Jennifer's mass * Jennifer's velocity
Initial momentum = 58.0 kg * 5.70 m/s
Now, the momentum of the system after Jennifer starts running should also be zero because there are no external forces acting on the system. The momentum of the system after Jennifer starts running can be calculated as:
Final momentum = (Jennifer's mass + Cart's mass) * Velocity of the system
Final momentum = (58.0 kg + 594.0 kg) * (Final velocity of the system)
Given that the final momentum is zero, we can substitute the known values to solve for the final velocity of the system:
(58.0 kg + 594.0 kg) * (Final velocity) = 0
Simplifying the equation:
Final velocity = - [(58.0 kg * 5.70 m/s) / 594.0 kg ]
Now, the distance Jennifer will have run relative to the ground is:
Distance = (Relative velocity of Jennifer) * (Time taken)
The relative velocity of Jennifer is her velocity relative to the cart plus the velocity of the system:
Relative velocity of Jennifer = Jennifer's velocity relative to the cart + Final velocity of the system
Relative velocity of Jennifer = 5.70 m/s + (Final velocity)
Substituting the calculated value of the final velocity, we can find the relative velocity of Jennifer.
Finally, substituting the known values, we can solve for the distance Jennifer will have run relative to the ground when she reaches the right end of the cart.