A truck with a mass of 2200 kg, moving at 31 m/s (69 mph), crashes into an immovable wall. It crumples; after the collision, the truck is 0.65m shorter in length than it was before.
What is the average speed and how long did the collision last?
I was confused on how to start out. There just seems to be so many variables and things changing.
I thought about average speed being the total distance over time, so I treated the 0.65m as the total distance traveled by the truck and divided that by it's initial velocity to see how long it would take for the car to cover that distance.
The answer I got was 2x10-2s. Is that right? Can I make those assumptions?
It just doesn't seem right since the velocity changed over time.
To solve this problem, we need to use the laws of motion and the principles of conservation of momentum and energy. Let's break it down step by step.
Step 1: Analyzing the problem and defining variables
- Mass of the truck (m) = 2200 kg
- Initial velocity of the truck (v_initial) = 31 m/s
- Final length of the truck (L_final) = L_initial - 0.65 m
Step 2: Calculate the change in velocity
Since the truck crashes into an immovable wall, its velocity changes to zero during the collision. We need to determine the change in velocity (Δv) using the conservation of momentum.
Since the wall does not move and has an infinite mass, the momentum before the collision is equal to the momentum after the collision:
Initial Momentum = Final Momentum
Initial Momentum: P_initial = m * v_initial
After the collision, the momentum is zero since the truck comes to a complete stop.
Final Momentum: P_final = 0
Therefore, we have:
Initial Momentum = Final Momentum
m * v_initial = 0
This implies that the change in velocity Δv = v_final - v_initial = 0 - v_initial = -v_initial.
Step 3: Calculate the average speed and collision duration
Now, we need to calculate the average speed and the duration of the collision.
Average Speed:
The average speed is the total distance traveled (Δx) divided by the total time taken (Δt).
Since the truck crumples and becomes shorter, we cannot directly use the change in length as the distance traveled during the collision. However, we know that the truck's acceleration during the collision is negative due to the deceleration caused by the impact.
Distance traveled (Δx) = (L_initial - L_final) / 2
Using the second equation of motion:
Δx = v_initial * Δt + (1/2) * (-a) * (Δt)^2
Given that the change in velocity (Δv) = -v_initial, we can substitute for acceleration (a = Δv/Δt) and rearrange the equation to solve for Δt:
(L_initial - L_final) / 2 = v_initial * Δt + (1/2) * (-Δv/Δt) * Δt^2
(L_initial - L_final) / 2 = v_initial * Δt - (1/2) * Δv * Δt
Simplifying the equation:
(Δt^2) * (Δv/2) - v_initial * Δt + (L_initial - L_final) / 2 = 0
We have a quadratic equation in terms of Δt. Solving this equation, we would get two values for Δt, one positive and one negative. We discard the negative value since time cannot be negative.
The positive value of Δt represents the duration of the collision.
Step 4: Calculate the final results
Now that we have our values, we can calculate:
- Average Speed: Average Speed = Δx / Δt
- Collision Duration: Δt (positive value from the quadratic equation)
Unfortunately, the values provided for the initial velocity, truck mass, and change in length are missing. Without these values, we cannot provide the final numerical answers you are seeking.
I apologize for the inconvenience. Please provide the missing information, and I'll be happy to help you solve the problem and get the accurate answers.