In a railroad yard, a 2.0×104 kg boxcar moving at 10m/s is brought to a stop by a spring-loaded bumper mounted at the end of the level track.
If the spring constant 1.7 MN/m , how far does it compress in stopping the boxcar?
What do i do?
To determine how far the spring compresses in stopping the boxcar, you can use the principle of conservation of mechanical energy. The initial kinetic energy of the boxcar will be converted into potential energy stored in the compressed spring.
Here are the steps to calculate the compression distance:
1. Determine the initial kinetic energy (KE_initial) of the boxcar using the formula: KE_initial = 0.5 * mass * velocity^2
Given:
mass (m) = 2.0 × 10^4 kg
velocity (v) = 10 m/s
Plugging in the values:
KE_initial = 0.5 * 2.0 × 10^4 kg * (10 m/s)^2
2. Calculate the potential energy (PE_spring) stored in the compressed spring using the formula: PE_spring = 0.5 * spring constant * compression^2
Given:
spring constant (k) = 1.7 MN/m
Unknown:
compression (x)
Plugging in the values:
PE_spring = 0.5 * 1.7 MN/m * (compression)^2
3. Apply the conservation of mechanical energy principle, which states that KE_initial = PE_spring.
So, equating the initial kinetic energy to the potential energy of the spring:
KE_initial = PE_spring
0.5 * 2.0 × 10^4 kg * (10 m/s)^2 = 0.5 * 1.7 MN/m * (compression)^2
4. Solve for compression (x) by rearranging the equation:
compression = √((2.0 × 10^4 kg * (10 m/s)^2) / (1.7 MN/m))
Simplifying the equation:
compression = √(2.0 × 10^4 kg * 100 m^2 / 1.7 × 10^6 N)
compression = √(2.0 × 10^2 m^2)
5. Calculate the compression distance:
compression = √(2.0 × 10^2) m
compression ≈ 14.142 m
Therefore, the spring compresses approximately 14.142 meters in stopping the boxcar.
To solve this problem, you can use the principle of conservation of mechanical energy. The initial kinetic energy of the boxcar will be converted into the potential energy stored in the compressed spring.
Here are the steps to find the compression distance of the spring:
1. Calculate the initial kinetic energy of the boxcar using the formula:
Kinetic Energy = 0.5 * mass * velocity^2
Substitute the given values:
Mass (m) = 2.0 × 10^4 kg
Velocity (v) = 10 m/s
Calculate the initial kinetic energy.
2. Calculate the potential energy stored in the spring using the formula:
Potential Energy = 0.5 * spring constant * compression distance^2
Substitute the given value of the spring constant:
Spring Constant (k) = 1.7 × 10^6 N/m
Assume the compression distance (x) as unknown and solve for it.
3. Equate the initial kinetic energy to the potential energy of the spring:
Initial Kinetic Energy = Potential Energy
Substitute the calculated values of the initial kinetic energy and the spring constant into this equation.
4. Solve for the unknown, compression distance (x), by rearranging the equation and isolating x.
Once you follow these steps, you will be able to determine the compression distance of the spring that stops the boxcar.
kinetic energy lost = (1/2) m v^2
potential energy gained = (1/2) k x^2