Since it is frictionless, use conservation of energy.
The work done equals the sum of kinetic and potential energy gains.
The work done equals the sum of kinetic and potential energy gains.
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.