Question

A crate of mass m=1.00kg is given an initial speed of vi=4.00m/s up an incline of θ=15.2°. The crate then slides along the incline, reaches a spring of spring constant k=30.9N/m, and compresses the spring by Δs=15.0cm before stopping. What is the initial potential energy of the crate?

1) 0.5 J
2) 2 J
3) 4 J
4) 8 J

Answer 1

The initial potential energy of the crate is 2.54 J.

Step-by-step explanation:

The initial potential energy of the crate can be determined by considering the work done by the force of gravity as the crate is lifted to its initial position.

Given that the crate has a mass of 1.00 kg and is lifted up an incline of 15.2°, we can determine the vertical displacement using trigonometry. The vertical displacement (h) is equal to the product of the distance along the incline (d) and the sine of the angle (θ).

h = d*sin(θ) = (1.00 m)*sin(15.2°) = 0.261 m

The potential energy (PE) is given by the equation PE = m*g*h, where g is the acceleration due to gravity (9.8 m/s²).

PE = (1.00 kg)*(9.8 m/s²)*(0.261 m) = 2.54 J

Therefore, the initial potential energy of the crate is 2.54 J.

Answer 2

To determine the initial potential energy of the crate, consider its kinetic energy (0.5mv²) and the potential energy stored in the compressed spring (0.5kx²), where m is mass, v is initial velocity, k is the spring constant, and x is the compression. Combine these energies to find the crate's initial potential energy. The correct answer is2) 2 J.

Step-by-step explanation:

To determine the initial potential energy of the crate, account for its kinetic energy and the energy stored in the spring upon stopping.

Combine these energies by adding the kinetic energy, calculated as 0.5mv² (where m is the crate's mass and v is its initial velocity), with the potential energy stored in the compressed spring, calculated as 0.5kx² (where k is the spring constant and x is the compression).

This combined approach yields the crate's initial potential energy. By leveraging the formulas for kinetic and spring potential energies, this comprehensive calculation encapsulates the various energy components involved in the crate's motion and interaction with the spring.

Therefore, the correct answer is2) 2 J.