A 2.9-kg cart is rolling along a frictionless, horizontal track towards a 1.5-kg cart that is held initially at rest. The carts are loaded with strong magnets that cause them to attract one another. Thus, the speed of each cart increases. At a certain instant before the carts collide, the first cart's velocity is +5.4 m/s, and the second cart's velocity is -1.4 m/s. (a) What is the total momentum of the system of the two carts at this instant? (b) What was the velocity of the first cart when the second cart was still at rest?
a. M1*V1 + M2*V2 = 2.9*5.4 - 1.5*1.4 = 13.6
To find the total momentum of the system of the two carts at the given instant, we can use the formula:
Total Momentum = Momentum of the First Cart + Momentum of the Second Cart
The momentum of an object is calculated by multiplying its mass by its velocity.
(a) The total momentum can be calculated as follows:
Momentum of the First Cart = mass of the first cart * velocity of the first cart
= 2.9 kg * 5.4 m/s
Momentum of the Second Cart = mass of the second cart * velocity of the second cart
= 1.5 kg * (-1.4 m/s)
Total Momentum = (2.9 kg * 5.4 m/s) + (1.5 kg * (-1.4 m/s))
= 15.66 kg*m/s + (-2.1 kg*m/s)
= 13.56 kg*m/s
Therefore, the total momentum of the system of the two carts at the given instant is 13.56 kg*m/s.
(b) To find the velocity of the first cart when the second cart was still at rest, we need to apply the Law of Conservation of Momentum. According to this law, the total momentum before the collision is equal to the total momentum after the collision.
Since the second cart is initially at rest, its initial velocity is 0 m/s. Let's assume the final velocity of both carts after the collision is v.
Using the same formula for momentum as before:
Momentum before collision = Momentum of the first cart + Momentum of the second cart
= (2.9 kg * 5.4 m/s) + (1.5 kg * 0 m/s)
= 15.66 kg*m/s
Total Momentum after collision = Momentum of the first cart + Momentum of the second cart
= (2.9 kg * v) + (1.5 kg * v)
= 4.4 kg * v
Since the total momentum is conserved, we can set the momentum before the collision equal to the momentum after the collision:
15.66 kg*m/s = 4.4 kg * v
Now, solve for v:
v = 15.66 kg*m/s / 4.4 kg
≈ 3.56 m/s
Therefore, the velocity of the first cart when the second cart was still at rest is approximately 3.56 m/s.
(a) The total momentum of a system is equal to the sum of the individual momenta of its objects. The momentum of an object is defined as the product of its mass and velocity.
To find the total momentum of the system, we calculate the momentum of each cart separately and then add them together.
Momentum of the first cart (m1):
mass of first cart (m1) = 2.9 kg
velocity of first cart (v1) = +5.4 m/s
Momentum of the first cart (p1) = m1 * v1
= 2.9 kg * 5.4 m/s
Momentum of the second cart (m2):
mass of second cart (m2) = 1.5 kg
velocity of second cart (v2) = -1.4 m/s
Momentum of the second cart (p2) = m2 * v2
= 1.5 kg * (-1.4 m/s)
Total momentum of the system = p1 + p2
(b) To find the velocity of the first cart when the second cart was still at rest, we can use the principle of conservation of momentum. According to this principle, the total momentum of a system before an event is equal to the total momentum after the event, assuming no external forces act on the system.
Here, the initial momentum of the system (before the carts collide) is equal to the final momentum of the system (after the carts collide), since no external forces are involved.
The initial momentum of the system = Total momentum before the carts collide = m1 * v1 + m2 * v2
The final momentum of the system = Total momentum after the carts collide = (m1 + m2) * v
Since the second cart is initially at rest (v2 = 0), we can substitute v2 = 0 in the equation for initial momentum:
Initial momentum of the system = m1 * v1 + m2 * v2
= (2.9 kg * 5.4 m/s) + (1.5 kg * 0 m/s)
The final momentum of the system remains the same as in part (a), which is (m1 + m2) * v:
Final momentum of the system = (m1 + m2) * v
= (2.9 kg + 1.5 kg) * v
Since the initial and final momenta are equal, we can equate these two expressions:
m1 * v1 + m2 * v2 = (m1 + m2) * v
Substituting the known values:
(2.9 kg * 5.4 m/s) + (1.5 kg * 0 m/s) = (2.9 kg + 1.5 kg) * v
Simplifying the equation and solving for v:
(2.9 kg * 5.4 m/s) = (4.4 kg) * v
v = (2.9 kg * 5.4 m/s) / (4.4 kg)
So, the velocity of the first cart when the second cart was still at rest is approximately +3.56 m/s (in the positive direction).