A skateboard track has the form of a circular arc with a 3.20 m radius, extending to an angle of 90.0° relative to the vertical on either side of the lowest point, as shown in Figure 8-24. A 67.0 kg skateboarder starts from rest at the top of the circular arc
OK, so it is a "half pipe".
What is the question?
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To find the speed of the skateboarder at the bottom of the circular arc, you can use the principle of conservation of energy. At the top of the arc, the skateboarder has potential energy due to their height, and at the bottom of the arc, this potential energy is converted into kinetic energy.
First, you need to find the potential energy of the skateboarder at the top of the arc. The potential energy formula is:
PE = mgh
where PE is the potential energy, m is the mass of the skateboarder (67.0 kg in this case), g is the acceleration due to gravity (9.8 m/s^2), and h is the height of the skateboarder at the top of the arc.
Since the skateboarder starts from rest at the top, all the potential energy is converted into kinetic energy at the bottom of the arc. The kinetic energy formula is:
KE = (1/2)mv^2
where KE is the kinetic energy and v is the velocity of the skateboarder at the bottom of the arc.
Using the conservation of energy, we can set the potential energy at the top equal to the kinetic energy at the bottom:
mgh = (1/2)mv^2
Here, mass (m) cancels out on both sides of the equation. Rearranging the equation, we get:
v = sqrt(2gh)
Now, substitute the given values into the equation to find the velocity (v) at the bottom of the arc:
v = sqrt(2 * 9.8 m/s^2 * 3.20 m)
v = sqrt(62.4) m/s
v ≈ 7.90 m/s
So, the skateboarder has a speed of approximately 7.90 m/s at the bottom of the circular arc.