4. A train car with mass m1 = 684 kg is moving to the right with a speed of v1 = 7.2 m/s and collides with a second train car. The two cars latch together during the collision and then move off to the right at vf = 4.4 m/s.
b) Now the same two cars are involved in a second collision. The first car is again moving to the right with a speed of v1 = 7.2 m/s and collides with the second train car that is now moving to the left with a velocity v2 = -5.4 m/s before the collision. The two cars latch together at impact.
What is the final velocity of the two-car system? (A positive velocity means the two train cars move to the right – a negative velocity means the two train cars move to the left.)
please explain with the answer
conservation of momentum applies
intital momentum = final
m1*V1+M2*V2= (M1+M2)Vf
put in the numbers (V2 to the left means -), solve for Vf
To find the final velocity of the two-car system after the second collision, we can use the principle of conservation of momentum. According to this principle, the total momentum before the collision is equal to the total momentum after the collision.
Let's denote the mass of the first train car as m1, the mass of the second train car as m2, the initial velocity of the first train car as v1, the initial velocity of the second train car as v2, and the final velocity of the two-car system as vf.
Before the second collision:
Total momentum = (mass of the first train car * initial velocity of the first train car) + (mass of the second train car * initial velocity of the second train car)
Total momentum = (m1 * v1) + (m2 * v2)
After the collision:
Total momentum = (mass of the two-car system * final velocity of the two-car system)
Total momentum = (m1 + m2) * vf
Since the two cars latch together at impact, the final velocity of the two-car system will be the same for both cars. Therefore, we can set the equations equal to each other:
(m1 * v1) + (m2 * v2) = (m1 + m2) * vf
Now we can plug in the given values:
m1 = 684 kg
v1 = 7.2 m/s
m2 = m1 (since it's the same two cars involved)
v2 = -5.4 m/s
(684 kg * 7.2 m/s) + (684 kg * (-5.4 m/s)) = (684 kg + 684 kg) * vf
Multiply and simplify the equation:
4910.4 kg·m/s - 3705.6 kg·m/s = 1368 kg * vf
1204.8 kg·m/s = 1368 kg * vf
Now we can solve for vf by dividing both sides of the equation by 1368 kg:
vf = 1204.8 kg·m/s / 1368 kg
vf ≈ 0.88 m/s
Therefore, the final velocity of the two-car system after the second collision is approximately 0.88 m/s to the right.
To find the final velocity of the two-car system after the second collision, you can use the principle of conservation of momentum. According to this principle, the total momentum of an isolated system remains constant before and after a collision.
In this case, the initial momentum of the first train car (m1) is given by:
p1 = m1 * v1
The initial momentum of the second train car (m2) is given by:
p2 = m2 * v2
At the moment of impact, the two cars latch together, so they become a single system. Let's assume the combined mass of the two cars is denoted by M.
The final velocity of the two-car system (vf) is given by:
vf = (p1 + p2) / M
Since we only have the mass and initial velocity of the first car (m1 and v1), and the initial velocity of the second car (v2), we need to determine the mass of the second car (m2) to calculate the final velocity.
To find m2, we can consider the principle of conservation of kinetic energy. In an elastic collision, the total kinetic energy of the system before and after the collision remains constant.
The initial kinetic energy of the system is given by:
KE_initial = (1/2) * m1 * v1^2 + (1/2) * m2 * v2^2
The final kinetic energy of the system is given by:
KE_final = (1/2) * M * vf^2
Since the two cars latch together, they have a common final velocity (vf) after the collision.
Setting the initial and final kinetic energies equal and solving for m2, we get:
(1/2) * m1 * v1^2 + (1/2) * m2 * v2^2 = (1/2) * M * vf^2
Now you can substitute the values given in the problem:
m1 = 684 kg
v1 = 7.2 m/s
v2 = -5.4 m/s
vf (which is what we want to find)
Solving the equation will give you the mass of the second car (m2). Once you have m2, you can substitute it back into the momentum equation to find the final velocity (vf) of the two-car system after the second collision.