An automobile having a mass of 1220 kg is driven into a brick wall in a safety test. The bumper behaves like a spring of constant 2.42 × 106 N/m and compresses 2.99 cm as the car is brought to rest. What was the speed of the car before im- pact, assuming no energy was lost during im- pact with the wall?
Energy=1/2 k x^2 solve for energy
KE = EPE
1/2m v^2 = 1/2k x^2
Make sure x is in meters.
To determine the speed of the car before the impact, we can start by calculating the total energy of the car before it hit the wall and then equate it to the energy stored in the compressed bumper spring.
1. Calculate the potential energy stored in the compressed bumper spring:
The potential energy (PE) stored in a spring is given by the equation: PE = (1/2) * k * x^2
where k is the spring constant and x is the compression.
Given:
Spring constant k = 2.42 × 10^6 N/m
Compression x = 2.99 cm = 0.0299 m
PE = (1/2) * (2.42 × 10^6 N/m) * (0.0299 m)^2
2. Calculate the total energy of the car before impact:
Since no energy is lost during the impact, the total energy of the car before impact is equal to the potential energy stored in the compressed bumper spring.
Total energy (E) = PE
3. Calculate the kinetic energy of the car using the total energy:
Kinetic energy (KE) is given by the equation: KE = (1/2) * m * v^2
where m is the mass of the car and v is the velocity.
Given:
Mass m = 1220 kg
Rearranging the equation, we get:
v^2 = (2 * E) / m
Substituting the value of E calculated in step 2, we can solve for v.
v^2 = (2 * PE) / m
v = sqrt((2 * PE) / m)
4. Solve for the velocity v:
Substitute the values of PE and m into the equation and calculate:
v = sqrt((2 * (1/2) * (2.42 × 10^6 N/m) * (0.0299 m)^2) / 1220 kg)
v = sqrt((2 * 1.44 * 10^5) / 1220)
v = sqrt(236.066)
v ≈ 15.36 m/s
Therefore, the speed of the car before the impact, assuming no energy was lost during the impact with the wall, is approximately 15.36 m/s.