A ball with a mass of 0.8 kg is kicked up the ramp with an initial speed of 22 m/s. What is the speed [m/s] of the ball at the top assuming no friction? Angle from which ball is kicked is 23 degrees and the horizontal leg is 7 feet long.
Hint: the elevation change of the ball is
H = 7 ft*0.3048 m/ft*tan 23 = 0.906 m
The PE gain is M g H
Subtract the PE gain from the initial KE to get the final KE. They use that for the speed at the top
To find the speed of the ball at the top of the ramp, we can use the principle of conservation of mechanical energy. In this scenario, there is no friction, so mechanical energy will be conserved.
The mechanical energy of the ball is the sum of its kinetic energy (KE) and potential energy (PE):
Mechanical Energy = KE + PE
At the bottom of the ramp, the entire mechanical energy is in the form of kinetic energy:
Mechanical Energy (bottom) = KE (bottom)
At the top of the ramp, the mechanical energy is in the form of both kinetic energy and potential energy:
Mechanical Energy (top) = KE (top) + PE (top)
Since there is no change in height, the potential energy at the top is equal to zero.
Therefore, we have:
Mechanical Energy (top) = KE (top) + 0
According to the conservation of mechanical energy, the mechanical energy at the bottom and the mechanical energy at the top are equal:
Mechanical Energy (bottom) = Mechanical Energy (top)
KE (bottom) = KE (top)
Now, let's calculate the initial kinetic energy at the bottom:
KE (bottom) = (1/2) * mass * velocity^2
mass = 0.8 kg
velocity = 22 m/s
KE (bottom) = (1/2) * 0.8 kg * (22 m/s)^2
KE (bottom) ≈ 211.2 J
Since kinetic energy is conserved, the kinetic energy at the top will also be 211.2 J:
KE (top) = 211.2 J
To find the speed of the ball at the top, we can rearrange the kinetic energy equation:
KE = (1/2) * mass * velocity^2
Let's solve for velocity:
velocity = √(2 * KE / mass)
velocity (top) = √(2 * 211.2 J / 0.8 kg)
velocity (top) ≈ √(528 J/kg)
velocity (top) ≈ 22.99 m/s
Therefore, the speed of the ball at the top of the ramp, assuming no friction, is approximately 22.99 m/s.