A delivery truck with a mass of 1700 kg is stopped in the street w/o its brakes on, when a small car hits it from behind. The care has a mass of 650 kg, and is traveling at 3.7m/s immediately before it hits the truck. Both the car and the truck have good bumpers, so assume the collision is elastic.
detailed solutioning with step by step instructions are greatly appreciated.
To solve this problem step-by-step, we can use the principles of conservation of momentum and kinetic energy in an elastic collision.
Step 1: Analyze the initial conditions
The delivery truck is initially at rest, so its initial velocity (before the collision) is 0 m/s. The small car has an initial velocity of 3.7 m/s.
Step 2: Find the final velocity of the car and truck
Since the collision is elastic, both the car and the truck will move after the collision.
Let's assume the final velocities of the car and the truck are v1 and v2, respectively.
Step 3: Apply the conservation of momentum
The total momentum before the collision should be equal to the total momentum after the collision.
Momentum before the collision = Momentum after the collision
(mass of the car) × (initial velocity of the car) = (mass of the car) × (final velocity of the car) + (mass of the truck) × (final velocity of the truck)
(650 kg) × (3.7 m/s) = (650 kg) × v1 + (1700 kg) × v2
Step 4: Apply the conservation of kinetic energy
Since the collision is elastic, the total kinetic energy before the collision should be equal to the total kinetic energy after the collision.
Kinetic energy before the collision = Kinetic energy after the collision
(0.5) × (mass of the car) × (initial velocity of the car)^2 = (0.5) × (mass of the car) × (final velocity of the car)^2 + (0.5) × (mass of the truck) × (final velocity of the truck)^2
(0.5) × (650 kg) × (3.7 m/s)^2 = (0.5) × (650 kg) × v1^2 + (0.5) × (1700 kg) × v2^2
Step 5: Solve the system of equations
Now we have two equations with two unknowns (v1 and v2). You can solve these equations using substitution or elimination.
Equation 1: (650 kg) × (3.7 m/s) = (650 kg) × v1 + (1700 kg) × v2
Equation 2: (0.5) × (650 kg) × (3.7 m/s)^2 = (0.5) × (650 kg) × v1^2 + (0.5) × (1700 kg) × v2^2
Solving these equations will give you the final velocities of the car and the truck after the collision.
Step 6: Evaluate the solution
Once you have found the values of v1 and v2, you can check if they make physical sense. For example, check if the final velocities are realistic and if they align with your expectations.
That's it! By following these steps, you can solve the problem and find the final velocities of the car and the truck after the collision.
To solve this problem, we need to use the principles of conservation of momentum and kinetic energy. Here are the step-by-step instructions to find the final velocities of the truck and car after the collision:
Step 1: Write down the given information:
- Mass of the truck (m1) = 1700 kg
- Mass of the car (m2) = 650 kg
- Initial velocity of the car before collision (v2i) = 3.7 m/s
Step 2: Remember the formula for momentum (P) and kinetic energy (KE):
- Momentum (P) = mass (m) × velocity (v)
- Kinetic energy (KE) = 0.5 × mass (m) × velocity^2 (v^2)
Step 3: Calculate the initial momenta (P) of the truck and the car before the collision:
- Initial momentum of the truck (P1i) = m1 × 0 (since the truck is stopped)
- Initial momentum of the car (P2i) = m2 × v2i
Step 4: Apply the principle of conservation of momentum (P1i + P2i = P1f + P2f) to find the final momenta of the truck and the car after the collision:
- Final momentum of the truck (P1f) = P1i
- Final momentum of the car (P2f) = P2i
Step 5: Calculate the final velocities (v1f and v2f) of the truck and the car using the final momenta and their respective masses:
- Final velocity of the truck (v1f) = P1f / m1
- Final velocity of the car (v2f) = P2f / m2
Step 6: Solve for the final velocities (v1f and v2f) using the above equations and the given values.
Let's work through the calculations:
Given:
- Mass of the truck (m1) = 1700 kg
- Mass of the car (m2) = 650 kg
- Initial velocity of the car before collision (v2i) = 3.7 m/s
Step 1: Given information.
Step 2: Equations for momentum and kinetic energy.
Step 3: Calculate the initial momenta.
- Initial momentum of the truck (P1i) = 1700 kg × 0 = 0 kg·m/s
- Initial momentum of the car (P2i) = 650 kg × 3.7 m/s = 2405 kg·m/s
Step 4: Apply the principle of conservation of momentum.
- Final momentum of the truck (P1f) = P1i = 0 kg·m/s
- Final momentum of the car (P2f) = P2i = 2405 kg·m/s
Step 5: Calculate the final velocities.
- Final velocity of the truck (v1f) = P1f / m1 = 0 kg·m/s / 1700 kg = 0 m/s
- Final velocity of the car (v2f) = P2f / m2 = 2405 kg·m/s / 650 kg = 3.7 m/s
Step 6: Final velocities.
- Final velocity of the truck (v1f) = 0 m/s
- Final velocity of the car (v2f) = 3.7 m/s
After the collision, the truck comes to a stop (v1f = 0 m/s), while the car maintains its initial velocity (v2f = 3.7 m/s).
Note: Since the collision is assumed to be elastic, kinetic energy is conserved. Thus, the total kinetic energy before the collision is equal to the total kinetic energy after the collision.