Car 1 has a mass of m1 = 65 103 kg and moves at a velocity of v01 = +0.78 m/s. Car 2, with a mass of m2 = 92 103 kg and a velocity of v02 = +1.4 m/s, overtakes car 1 and couples to it. Neglect the effects of friction in your answer.
(a) Determine the velocity of their center of mass before the collision
m/s
(b) Determine the velocity of their center of mass after the collision
m/s
(c) Should your answer in part (b) be less than, greater than, or equal to the common velocity vf of the two coupled cars after the collision?
c) they should be the same, momentum is conserved.
a) M1V1+M2V2=(M1+M2)Vcmass
this answers b) also.
To solve this problem, we 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.
(a) The velocity of the center of mass before the collision can be determined using the formula:
v_cm = (m1*v01 + m2*v02) / (m1 + m2)
Substituting the given values:
v_cm = (65 * 10^3 kg * 0.78 m/s + 92 * 10^3 kg * 1.4 m/s) / (65 * 10^3 kg + 92 * 10^3 kg)
Calculating this expression:
v_cm = (50700 kg*m/s + 128800 kg*m/s) / (157000 kg)
v_cm = 179500 kg*m/s / 157000 kg
v_cm ≈ 1.143 m/s
Therefore, the velocity of their center of mass before the collision is approximately 1.143 m/s.
(b) The velocity of the center of mass after the collision can be determined using the same formula as in part (a):
v'_cm = (m1*v'1 + m2*v'2) / (m1 + m2)
Since the cars are coupled and move together after the collision, their velocities will be the same. Let's denote this common velocity by vf.
v'1 = vf and v'2 = vf
Substituting these values into the formula, we get:
v'_cm = (m1*vf + m2*vf) / (m1 + m2)
Simplifying this expression:
v'_cm = vf * (m1 + m2) / (m1 + m2)
Canceling out the common factors:
v'_cm = vf
Therefore, the velocity of their center of mass after the collision is equal to the common velocity, vf.
(c) Based on the previous analysis, the answer in part (b) is equal to the common velocity, vf.
To answer this question, we 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.
(a) To determine the velocity of their center of mass before the collision, we need to calculate the total momentum of the system before the collision. The formula for momentum is given by:
Momentum = mass x velocity
For car 1:
Momentum1 = m1 x v01 = (65 103 kg) x (+0.78 m/s)
For car 2:
Momentum2 = m2 x v02 = (92 103 kg) x (+1.4 m/s)
The total momentum before the collision is:
Total momentum = Momentum1 + Momentum2
(b) To determine the velocity of their center of mass after the collision, we need to calculate the total mass of the system and the total momentum after the collision. The formula for the center of mass velocity is given by:
Center of mass velocity = Total momentum / Total mass
The total mass of the system after the collision is the sum of the masses of car 1 and car 2.
Total mass = m1 + m2
To find the total momentum after the collision, we need to consider that the two cars will move together as one unit. Since no external forces are acting on the system, the total momentum after the collision will be equal to the total momentum before the collision.
Total momentum after collision = Total momentum before collision
(c) The center of mass velocity after the collision can be compared to the common velocity vf of the two coupled cars. If the center of mass velocity is less than vf, it means the two cars are slowing down after the collision and the answer in part (b) will be less than vf. If the center of mass velocity is greater than vf, it means the two cars are speeding up after the collision and the answer in part (b) will be greater than vf. If the center of mass velocity is equal to vf, it means the two cars maintain the same velocity after the collision.