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?
To find the final speed and direction of the three vehicles after the collision, we need to apply the concepts of conservation of momentum. The total momentum before the collision must be equal to the total momentum after the collision.
Momentum is defined as the product of mass and velocity. The momentum of an object can be calculated using the formula:
Momentum = mass × velocity
Let's assign some variables to the given information:
Truck mass = 2050 kg, Truck initial velocity = 2.3 m/s (east)
Compact car mass = 1200 kg, Compact car initial velocity = 4.5 m/s (north)
Midsize car mass = 1600 kg, Midsize car initial velocity = 10 m/s (east)
First, we need to find the initial momentum of each vehicle in both the x and y directions:
Truck momentum in the x direction = Truck mass × Truck initial velocity
= 2050 kg × 2.3 m/s
Compact car momentum in the y direction = Compact car mass × Compact car initial velocity
= 1200 kg × 4.5 m/s
Midsize car momentum in the x direction = Midsize car mass × Midsize car initial velocity
= 1600 kg × 10 m/s
Since the compact car is traveling in the y direction and the truck and midsize car are traveling in the x direction, the compact car's momentum does not contribute to the final momentum in the x direction and vice versa.
The total initial momentum in the x direction is the sum of the truck and midsize car's momenta:
Total initial momentum in the x direction = Truck momentum in the x direction + Midsize car momentum in the x direction
Next, knowing that momentum is conserved and that all three vehicles become entangled and slide as one body, we assume that the final momentum in the x direction is equal to the initial momentum in the x direction.
Hence, the total initial momentum in the x direction = total final momentum in the x direction
Using this concept, we can calculate the final momentum of the three entangled vehicles in the x direction.
Finally, we can use the final momentum in the x direction to find the final velocity of the entangled vehicles. We divide the final momentum in the x direction by the combined mass of the three vehicles.
To determine the direction, we consider that the final velocity will be positive if the entangled vehicles move in the eastward direction and negative if they move in the westward direction.
The final speed and direction can be calculated by performing the above steps.