To find the speed at which the cyclist crests the hill, we need to apply the conservation of energy principle. According to this principle, the initial kinetic energy of the cyclist must be equal to or greater than the potential energy change at the top of the hill.
Let's break down the problem step-by-step:
Step 1: Given values
- Initial speed (Vi) = 20 m/s
- Height of the hill (H) = 4.4 m
- Acceleration due to gravity (g) = 9.8 m/s^2
Step 2: Calculate the potential energy change
The potential energy change is given by the equation: PE = mgh, where m is the mass of the cyclist, g is the acceleration due to gravity, and h is the height of the hill. However, since mass (m) cancels out in both kinetic and potential energy equations, we can ignore it in our calculations.
Potential energy change (PE) = gh
PE = (9.8 m/s^2) * (4.4 m)
PE = 43.12 J
Step 3: Determine the minimum required initial kinetic energy
Since the cyclist is coasting without pedaling, the only energy available is her initial kinetic energy.
Initial kinetic energy (KE) = (1/2)mv^2
KE = (1/2) * v^2
To crest the hill, the initial kinetic energy must be equal to or greater than the potential energy change. So, we have:
KE >= PE
(1/2) * v^2 >= 43.12 J
Step 4: Solve for the minimum required speed
To find the speed at which the cyclist crests the hill, we need to solve the above equation for v:
(1/2) * v^2 >= 43.12 J
v^2 >= 86.24 J
v >= sqrt(86.24 J)
v >= 9.29 m/s
Therefore, the cyclist must have a speed greater than or equal to 9.29 m/s to crest the hill without pedaling.
If this answer does not match the one you obtained, please verify your calculations and double-check the given values to see if any mistakes were made.