Lauren is pedaling down a steep hill on her bicycle and wants to be able to coast up the next hill, which is 25 meters high, without pedaling up the hill at all. What would her speed at the bottom need to be?
50 m/s
71 m/s
7.1 m/s
240 m/s
To determine the speed Lauren would need at the bottom of the hill to coast up the next hill without pedaling, we can use the principle of conservation of energy.
The potential energy (PE) at the top of the hill is converted to kinetic energy (KE) as Lauren rides down the hill. At the bottom of the hill, the KE is then converted back to potential energy to help her coast up the next hill.
The equation for potential energy is:
PE = m * g * h
Where:
m is the mass of Lauren and her bicycle
g is the acceleration due to gravity (approximately 9.8 m/s^2)
h is the height of the hill (25 m)
Since KE = 1/2 * m * v^2, where v is the velocity (speed) at the bottom of the hill, we can equate the potential energy at the top to the kinetic energy at the bottom:
m * g * h = 1/2 * m * v^2
Canceling out the mass term on both sides, we get:
g * h = 1/2 * v^2
Now, we can solve for v by rearranging the equation:
v^2 = 2 * g * h
v = √(2 * g * h)
Plugging in the values, we have:
v = √(2 * 9.8 * 25)
v = √(2 * 245)
v = √(490)
v ≈ 22.14 m/s
Therefore, Lauren would need a speed of approximately 22.14 m/s at the bottom of the hill to be able to coast up the next hill without pedaling. Since none of the given answer choices match this value, none of the provided options are correct.