A 72.0 kg skateboarder wants to just make it to the upper edge of a "quarter pipe", a track that is one-quarter of a circle with a radius of 4.65 m. What speed (in m/s) does he need at the bottom assuming that the local acceleration due to gravity is 9.80 m/s2?
5.88m/s
To find the speed that the skateboarder needs at the bottom of the quarter pipe, we can use the conservation of mechanical energy.
The potential energy at the bottom is given by the formula:
PE = mgh
where m is the mass of the skateboarder, g is the acceleration due to gravity, and h is the height of the quarter pipe.
The potential energy at the top of the quarter pipe is given by:
PE = mgh'
where h' is the height at the upper edge of the quarter pipe.
The initial kinetic energy at the bottom is given by:
KE = (1/2)mv^2
where v is the speed of the skateboarder at the bottom.
At the top of the quarter pipe, all the energy is in the form of potential energy, so we can equate the two expressions for potential energy:
mgh = mgh'
Since the mass of the skateboarder cancels out, we have:
gh = gh'
We can solve for h' in terms of h:
h' = h
Now we can equate the expressions for potential and kinetic energy:
mgh = (1/2)mv^2
Canceling out the mass and solving for v gives:
v^2 = 2gh
Taking the square root of both sides gives:
v = sqrt(2gh)
Substituting the given values:
v = sqrt(2 * 9.80 m/s^2 * 4.65 m)
v ≈ 9.61 m/s
Therefore, the skateboarder needs a speed of approximately 9.61 m/s at the bottom to just make it to the upper edge of the quarter pipe.
To find the speed the skateboarder needs at the bottom of the quarter pipe, we can use the principle of conservation of energy.
At the bottom of the quarter pipe, the skateboarder has both potential energy and kinetic energy. At the topmost point, all the potential energy is transformed into kinetic energy.
The potential energy (PE) at the top can be calculated using the formula:
PE = m * g * h
Where:
m = mass of the skateboarder = 72.0 kg
g = acceleration due to gravity = 9.80 m/s²
h = height from the bottom to the topmost point of the quarter pipe
Since the quarter pipe is a track of one-quarter of a circle with a radius of 4.65 m, the height of the quarter pipe can be calculated as follows:
h = radius of the quarter pipe = 4.65 m
Thus, the potential energy at the top is:
PE = 72.0 kg * 9.80 m/s² * 4.65 m
Next, at the bottom of the quarter pipe, all of this potential energy is transformed into kinetic energy (KE). The kinetic energy can be calculated using the formula:
KE = (1/2) * m * v²
Where:
m = mass of skateboarder = 72.0 kg
v = velocity of the skateboarder at the bottom
Since the potential energy at the top is equal to the kinetic energy at the bottom, we can set up an equation:
PE = KE
m * g * h = (1/2) * m * v²
Simplifying the equation, we find:
g * h = (1/2) * v²
Now, we can solve for v:
v² = 2 * g * h
v = √(2 * g * h)
Plugging in the values we know:
v = √(2 * 9.80 m/s² * 4.65 m)
Calculating this, we find:
v ≈ 9.19 m/s
Therefore, the skateboarder needs a speed of approximately 9.19 m/s at the bottom to make it to the upper edge of the quarter pipe.