A 1300-kg car moving north at 27.0 m/s is struck by a 2165-kg car moving east at 17.0 m/s. The cars are stuck together. How fast and in what direction do they move immediately after the collision?
X = m1*V1 = 2165 * 17 = 36,805
Y = m2*V2 = 1300 * 27 = 35,100
Tan A = Y/X = 35,100/36,805 = 0.95367
A = 43.64o
m*V = X/Cos A = 36,805/Cos43.64 = 50,859
(m1+m2)V = 50,859
(2165+1300)V = 50,859
V = 50,859/(2165+1300) = 14.7 m/s. @ 43.64o
To solve this problem, we can apply the conservation of momentum. The total momentum before the collision is equal to the total momentum after the collision.
1. Find the initial momentum of each car:
- Momentum (p) = mass (m) × velocity (v)
For the first car (car 1):
Mass (m1) = 1300 kg
Velocity (v1) = 27.0 m/s
Momentum (p1) = m1 × v1
For the second car (car 2):
Mass (m2) = 2165 kg
Velocity (v2) = 17.0 m/s
Momentum (p2) = m2 × v2
2. Calculate the total initial momentum:
Total initial momentum = p1 + p2
3. Determine the direction of the initial momentum:
Since car 1 is moving north and car 2 is moving east, the initial momentum is a combination of north and east components. To simplify calculations, we can break down the initial momentum into its north and east components.
To find the north component of momentum, use the formula:
p1_north = m1 × v1_north
where v1_north = v1 (since car 1 is already moving north)
To find the east component of momentum, use the formula:
p2_east = m2 × v2_east
where v2_east = v2 (since car 2 is already moving east)
Total initial north component momentum = p1_north
Total initial east component momentum = p2_east
4. Determine the total initial momentum magnitude and direction:
To find the magnitude of the initial momentum, use the Pythagorean theorem:
Total initial momentum magnitude = sqrt((Total initial north component momentum)^2 + (Total initial east component momentum)^2)
To find the direction of the initial momentum, use trigonometry:
Initial momentum angle (θ) = atan((Total initial north component momentum) / (Total initial east component momentum))
5. Calculate the final momentum after the collision:
The final momentum will be the same as the initial momentum since the cars stick together.
6. Determine the magnitude and direction of the final momentum:
The magnitude of the final momentum is equal to the magnitude of the initial momentum.
The direction of the final momentum is the same as the direction of the initial momentum.
To summarize:
- Calculate the initial momentum of each car using mass and velocity.
- Determine the north and east components of the initial momentum.
- Calculate the total initial momentum magnitude and direction.
- The final momentum will be the same as the initial momentum.
- The magnitude of the final momentum is equal to the magnitude of the initial momentum, and the direction is the same as the initial momentum direction.