A cyclist approaches the bottom of a gradual hill at a speed of 20 m/s. The hill is 4.4 m high, and the cyclist estimates that she is going fast enough to coast up and over it without peddling. Ignoring air resistance and friction, find the speed at which the cyclist crests the hill.
I posted this problem earlier and it was replied to with : The initial kinetic energy (1/2)M V^2 must equal or exceed the potential energy change at the top of the "hill", which is M g H.
Therefore V > sqrt (2 g H)
I was not able to find the correct answer.
KE at base of hill gets converted into potential energy at the top, plus remaining kinetic energy of bicycle moving. Thus, KE(base) = PE(top) + KE(top).
(mv^2)/2 base = mgh + (mv^2)/2 top.
Note the masses cancel out.
v^2/2(base) = gh + v^2/2(top)
400/2 = (9.8)(4.4) + v^2/2(top).
So you solve for v(top). This is the speed at which she is traveling over the top of the hill.
I apologize for misreading your question.
I forgot that there is an initial velocity of 20 m/s. I thought they were asking for the minimum velocity to coast to the top of the hill.
The velocity decreases from V1 to V2 while coasting uphill so that the change in kinetic energy equals the potential energy increase
( M/2)(V1^2 - V2^2) = M g H
V2^2 = V1^2 - 2 g H = 400 - 2*9.8*4.4
= 400 - 86.24 = 313.76
V2 = 17.7 m/s is the velocity at the top of the hill.
A cyclist approaches the bottom of a gradual hill at a speed of 15 m/s. The hill is 4.7 m high, and the cyclist estimates that she is going fast enough to coast up and over it without peddling. Ignoring air resistance and friction, find the speed at which the cyclist crests the hill.
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.