On an icy road, a 2100-kg car moving at 52 km/h strikes a 410-kg truck moving in the same direction at 35 km/h. The pair is soon hit from behind by a 1800-kg car speeding at 45 km/h, and all three vehicles stick together. Find the speed of the wreckage.
Given: M1 = 2100 kg, V1 = 52 km/h.
M2 = 410 kg, V2 = 35km/h.
M3 = 1800 kg, V3 = 45 km/h.
V4 = velocity of M1 and M2 after collision.
V5 = velocity of M1, M2, and M3 after collision.
Momentum before = Momentum after.
M1*V1 + M2*V2 = M1*V4 + M2*V4.
2100*52 + 410*35 = 2100*V4 + 410*V4,
2510V4 = 123,550,
V4 = 49.22 km/h.
M3 Joins Wreckage:
M1*V4 + M2*V4 + M3*V3 = M1*V5 + M2*V5 + M3*V5.
2100*49.22 + 410*49.22 + 1800*45 = 2100V5 + 410V5 + 1800V5,
4310V5 = 204,542,
V5 = 47.5 km/h.
To find the speed of the wreckage, we need to use the principle of conservation of momentum, which states that the total momentum before the collision is equal to the total momentum after the collision.
The momentum of an object can be calculated by multiplying its mass by its velocity. Therefore, the total momentum before the collision is:
Momentum of the 2100-kg car = (mass of the car) * (velocity of the car)
= 2100 kg * 52 km/h
Momentum of the 410-kg truck = (mass of the truck) * (velocity of the truck)
= 410 kg * 35 km/h
Momentum of the 1800-kg car = (mass of the car) * (velocity of the car)
= 1800 kg * 45 km/h
Now, let's calculate the individual momenta:
- The momentum of the 2100-kg car is equal to (2100 kg * 52 km/h).
- The momentum of the 410-kg truck is equal to (410 kg * 35 km/h).
- The momentum of the 1800-kg car is equal to (1800 kg * 45 km/h).
Next, let's find the total momentum before the collision by adding up these individual momenta.
Total momentum before the collision = (momentum of the 2100-kg car) + (momentum of the 410-kg truck) + (momentum of the 1800-kg car)
Now, we divide the total momentum before the collision by the total mass of the wreckage (sum of the masses of the three vehicles) to find the velocity of the wreckage. The total mass of the wreckage is (2100 kg + 410 kg + 1800 kg).
Finally, convert the velocity of the wreckage from km/h to m/s by dividing it by 3.6 (since 1 km/h is equal to 1/3.6 m/s).
Therefore, by calculating the above steps, we can find the speed of the wreckage.
To find the speed of the wreckage, we need to apply the principles of conservation of momentum.
Step 1: Convert the velocities from km/h to m/s.
- The velocity of the first car is 52 km/h, which is equivalent to 52 * (1000/3600) m/s = 14.44 m/s.
- The velocity of the truck is 35 km/h, which is equivalent to 35 * (1000/3600) m/s = 9.72 m/s.
- The velocity of the third car is 45 km/h, which is equivalent to 45 * (1000/3600) m/s = 12.50 m/s.
Step 2: Calculate the total momentum before the collision.
- The momentum of the first car is 2100 kg * 14.44 m/s = 30284.4 kg⋅m/s.
- The momentum of the truck is 410 kg * 9.72 m/s = 3985.2 kg⋅m/s.
- The momentum of the third car is 1800 kg * 12.50 m/s = 22500 kg⋅m/s.
The total momentum before the collision is the sum of these individual momenta:
Total momentum before = Momentum of first car + Momentum of truck + Momentum of third car
Total momentum before = 30284.4 kg⋅m/s + 3985.2 kg⋅m/s + 22500 kg⋅m/s
Step 3: Calculate the total mass of the wreckage.
The total mass of the wreckage is the sum of the masses of the three vehicles:
Total mass = Mass of first car + Mass of truck + Mass of third car
Total mass = 2100 kg + 410 kg + 1800 kg
Step 4: Calculate the speed of the wreckage.
- Apply the principle of conservation of momentum, which states that the total momentum before the collision is equal to the total momentum after the collision.
- Assume the speed of the wreckage after the collision is V m/s.
- The total momentum after the collision would be the product of the total mass and the speed of the wreckage.
Total momentum after = Total mass of the wreckage * V
Total momentum after = (2100 kg + 410 kg + 1800 kg) * V
Since the total momentum before the collision is equal to the total momentum after the collision, we can equate the two expressions:
30284.4 kg⋅m/s + 3985.2 kg⋅m/s + 22500 kg⋅m/s = (2100 kg + 410 kg + 1800 kg) * V
Solving for V will give us the speed of the wreckage after the collision.