A mass of 18 kg is being pushed up a frictionless incline with a constant force of 20 Newtons directed parallel to the incline, up the incline. At the top of the incline the mass is moving at 23 m/s up the incline. If the angle of inclination is 58 degrees and the height (not length) of the incline is 24 meters, what was the magnitude of the mass's velocity at the bottom of the incline in m/s?
Since it is frictionless, use conservation of energy.
The work done equals the sum of kinetic and potential energy gains.
To find the magnitude of the mass's velocity at the bottom of the incline, we need to apply the principle of conservation of energy. The initial potential energy of the mass at the top of the incline is equal to the final kinetic energy of the mass at the bottom of the incline.
Step 1: Determine the initial potential energy (PEi) at the top of the incline.
The formula for potential energy is PE = mgh, where m is the mass, g is the acceleration due to gravity, and h is the height.
PEi = 18 kg * 9.8 m/s^2 * 24 m
Step 2: Determine the final kinetic energy (KEf) at the bottom of the incline.
The formula for kinetic energy is KE = 1/2 * mv^2, where m is the mass and v is the velocity.
KEf = 1/2 * 18 kg * (v^2)
Step 3: Set the initial potential energy equal to the final kinetic energy.
PEi = KEf
18 kg * 9.8 m/s^2 * 24 m = 1/2 * 18 kg * (v^2)
Step 4: Solve for the magnitude of the velocity (v).
Using the given values, the equation becomes:
18 kg * 9.8 m/s^2 * 24 m = 1/2 * 18 kg * (v^2)
v^2 = (35.28 m^2/s^2 * 24 m) / 9 kg
v^2 = 94.08 m^2/s^2
To find the magnitude of the velocity, take the square root of both sides:
v = √(94.08 m^2/s^2)
v ≈ 9.70 m/s
Therefore, the magnitude of the mass's velocity at the bottom of the incline is approximately 9.70 m/s.