Two cars approach an ice-covered intersection. One car, of mass 1.22 103 kg, is initially traveling north at 10.6 m/s. The other car, of mass 1.69 103 kg, is initially traveling east at 10.6 m/s. The cars reach the intersection at the same instant, collide, and move off coupled together. Find the velocity of the center of mass of the two-car system just after the collision
To find the velocity of the center of mass of the two-car system just after the collision, we can use the principle of conservation of momentum. According to this principle, the total momentum of a system remains constant if no external forces act upon it.
The momentum of an object is calculated by multiplying its mass with its velocity. Therefore, total momentum before the collision is equal to the total momentum after the collision.
Let's denote the velocity of the center of mass of the two-car system just after the collision as V_cm. We can calculate the total momentum before the collision, ΣPi, by summing the individual momenta of the two cars:
ΣPi = m1 × v1 + m2 × v2
Where:
m1 and m2 are the masses of the two cars (1.22 x 10^3 kg and 1.69 x 10^3 kg, respectively)
v1 and v2 are the velocities of the two cars before the collision (10.6 m/s north and 10.6 m/s east, respectively)
ΣPi = (1.22 × 10^3 kg) × (10.6 m/s) + (1.69 × 10^3 kg) × (10.6 m/s)
Once we calculate the total momentum before the collision, we can use it to determine the total momentum after the collision:
ΣPf = (m1 + m2) × V_cm
Where:
ΣPf is the total momentum after the collision
m1 and m2 are the masses of the two cars
V_cm is the velocity of the center of mass of the two-car system just after the collision
Since momentum is conserved, we can set these two equations equal to each other:
ΣPi = ΣPf
(m1 × v1 + m2 × v2) = (m1 + m2) × V_cm
Now we can solve this equation to find the velocity of the center of mass (V_cm).