a skateboard with a mass of 1.5 kg moves up a quarter pipe to a height of 5.0 meters. How fast was it moving at ground level?
since all KE was converted to PE,
1/2 mv^2 = mgh
v^2 = 2gh
Now just plug in the numbers.
To find the initial velocity of the skateboard at ground level, we can use the principle of conservation of mechanical energy.
The total mechanical energy of the skateboard at ground level is equal to the total mechanical energy of the skateboard at the top of the quarter pipe.
The total mechanical energy (E) of an object can be calculated using the formula:
E = KE + PE
Where:
KE = kinetic energy
PE = potential energy
At ground level, the skateboard has only kinetic energy since it hasn't gained any potential energy yet. Therefore, the total mechanical energy at ground level is equal to the kinetic energy.
The formula for kinetic energy (KE) is:
KE = (1/2) * m * v^2
Where:
m = mass of the skateboard
v = velocity of the skateboard
Let's substitute the given values into the formula:
m = 1.5 kg (given)
v = ?
At the top of the quarter pipe, the skateboard reaches a height of 5.0 meters, so its potential energy (PE) is given by:
PE = m * g * h
Where:
g = gravitational acceleration (approximately 9.8 m/s^2)
h = height (5.0 meters in this case)
Let's calculate the potential energy at the top of the quarter pipe:
PE = 1.5 kg * 9.8 m/s^2 * 5.0 m
PE = 73.5 J
Since the total mechanical energy is conserved, the kinetic energy at ground level is equal to the potential energy at the top of the quarter pipe:
KE = 73.5 J
Now, we can solve for the velocity (v) using the formula for kinetic energy:
KE = (1/2) * m * v^2
73.5 J = (1/2) * 1.5 kg * v^2
Now, rearrange the equation to solve for v:
v^2 = (2 * 73.5 J) / (1.5 kg)
v^2 = 98 J/kg
v ≈ √98 m/s
v ≈ 9.9 m/s
Therefore, the skateboard was moving at approximately 9.9 m/s at ground level.