A 61 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 2.89 m. What speed does he need at the bottom?
To find the speed that the skateboarder needs at the bottom to reach the upper edge of the quarter pipe, we can use the principle of conservation of energy.
The potential energy at the bottom of the quarter pipe is transformed into both kinetic energy and gravitational potential energy at the upper edge. The total mechanical energy (E) at any point remains constant.
Let's break down the steps to find the required speed:
Step 1: Calculate the potential energy at the bottom.
At the bottom of the quarter pipe, all of the skateboarder's 61 kg mass is at the height of the radius of the quarter pipe, which is 2.89 m. The potential energy (PE) can be calculated using the formula:
PE = m * g * h,
where m is the mass (61 kg), g is the acceleration due to gravity (9.8 m/s²), and h is the height (2.89 m).
PE = 61 kg * 9.8 m/s² * 2.89 m
Step 2: Calculate the speed at the upper edge using the conservation of energy.
The total energy at the upper edge of the quarter pipe is the sum of the skateboarder's initial kinetic energy (KE) at the bottom and the gravitational potential energy (PE) at the top. When the skateboarder reaches the upper edge, the potential energy is zero as the skateboarder is at the highest point. The total energy (E) becomes:
E = KE + PE
Since E remains constant, we can equate the potential energy at the bottom (PE) to the initial kinetic energy (KE) at the top.
Potential energy at the bottom (PE) = Kinetic energy at the top (KE)
m * g * h = (1/2) * m * v²,
where v is the required velocity.
Step 3: Solve for the velocity.
Let's rearrange the equation and solve for v:
v = √(2 * g * h),
where √ stands for the square root.
v = √(2 * 9.8 m/s² * 2.89 m)
By substituting the given values and calculating the equation, you can find the required velocity for the skateboarder at the bottom to reach the upper edge of the quarter pipe.