A 2050 kg truck is traveling east through an intersection at 2.3 m/s when it is hit simultaneously from the side and the rear. (Some people have all the luck!) One car is a 1200 kg compact traveling north at 4.5 m/s. The other is a 1600 kg midsize traveling east at 10 m/s. The three vehicles become entangled and slide as one body. What are their speed and direction just after the collision?
conservation of momentum says that the result is
2050(2.3i+0j)+1200(0i+4.5j)+1600(10i+j) = (2050+1200+1600)(ai+bj)
4715i+5400j+16000i = 4850(ai+bj)
20715i+5400j = 4850(ai+bj)
4.27i+1.11j = ai+bj
speed is 4.42 m/s
direction E14.5°N
To determine the speed and direction of the vehicles just after the collision, we need to apply the principle of conservation of momentum. According to this principle, the total momentum of an isolated system remains constant before and after a collision.
Let's analyze the situation step by step:
Step 1: Convert the masses and velocities of the vehicles into vector form:
Truck: Mass (m1) = 2050 kg, Velocity (v1) = 2.3 m/s (east)
Compact: Mass (m2) = 1200 kg, Velocity (v2) = 4.5 m/s (north)
Midsize: Mass (m3) = 1600 kg, Velocity (v3) = 10 m/s (east)
Step 2: Calculate the initial momentum of each vehicle:
Initial momentum of truck (p1) = m1 * v1 = 2050 kg * 2.3 m/s (east)
Initial momentum of compact (p2) = m2 * v2 = 1200 kg * 4.5 m/s (north)
Initial momentum of midsize (p3) = m3 * v3 = 1600 kg * 10 m/s (east)
Step 3: Resolve the velocities into their x and y components:
v1x = 2.3 * cos(0°) = 2.3 m/s (east)
v2x = 4.5 * cos(90°) = 0 m/s
v3x = 10 * cos(0°) = 10 m/s (east)
v1y = 2.3 * sin(0°) = 0 m/s
v2y = 4.5 * sin(90°) = 4.5 m/s (north)
v3y = 10 * sin(0°) = 0 m/s
Step 4: Calculate the total initial momentum of the system in the x and y directions:
Initial momentum in x-direction = p1x + p2x + p3x
Initial momentum in y-direction = p1y + p2y + p3y
Step 5: Determine the final momentum of the system in the x and y directions:
Since all three vehicles become entangled and slide together, their final x and y velocities will be the same.
Final momentum in x-direction = (m1 + m2 + m3) * final velocity in x-direction
Final momentum in y-direction = (m1 + m2 + m3) * final velocity in y-direction
Step 6: Apply the conservation of momentum principle to solve for the final velocities:
Initial momentum in x-direction = Final momentum in x-direction
Initial momentum in y-direction = Final momentum in y-direction
We can now solve the system of equations to determine the final velocities of the entangled vehicles.
Note: Since the truck is hit simultaneously from the side and the rear, we can assume that the collision is perfectly inelastic, meaning the vehicles stick together after the collision.
By applying the conservation of momentum, you can find the final velocities of the system. I will perform the calculations for you.
Initial momentum in x-direction = p1x + p2x + p3x
Initial momentum in y-direction = p1y + p2y + p3y
p1x = 2050 kg * 2.3 m/s = 4715 kg·m/s (east)
p2x = 0 kg·m/s (Since the compact is moving north, it has no momentum in the x-direction)
p3x = 1600 kg * 10 m/s = 16000 kg·m/s (east)
p1y = 0 kg·m/s (The truck is moving east, so it has no momentum in the y-direction)
p2y = 1200 kg * 4.5 m/s = 5400 kg·m/s (north)
p3y = 0 kg·m/s (The midsize is moving east, so it has no momentum in the y-direction)
Initial momentum in x-direction = 4715 kg·m/s + 0 kg·m/s + 16000 kg·m/s = 20715 kg·m/s (east)
Initial momentum in y-direction = 0 kg·m/s + 5400 kg·m/s + 0 kg·m/s = 5400 kg·m/s (north)
Since the collision is perfectly inelastic, the final velocity of the system will be the same in both x and y directions.
Final momentum in x-direction = (m1 + m2 + m3) * final velocity in x-direction
Final momentum in y-direction = (m1 + m2 + m3) * final velocity in y-direction
Final momentum in x-direction = 20715 kg·m/s + 0 kg·m/s + 0 kg·m/s = 20715 kg·m/s
Final momentum in y-direction = 0 kg·m/s + 5400 kg·m/s + 0 kg·m/s = 5400 kg·m/s
The final velocity of the system will be the same in both x and y directions:
Final momentum in x-direction = (m1 + m2 + m3) * final velocity in x-direction
20715 kg·m/s = (2050 kg + 1200 kg + 1600 kg) * final velocity in x-direction
20715 kg·m/s = 4850 kg * final velocity in x-direction
final velocity in x-direction = 20715 kg·m/s / 4850 kg ≈ 4.27 m/s
Therefore, the speed and direction of the vehicles just after the collision is approximately 4.27 m/s to the east.