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
PE top of next hill=mg*25
KE she must have at bottom=1/2 m v^2
set them equal, solve for v.
To find out Lauren's required speed at the bottom of the hill, we can use the principle of conservation of mechanical energy. According to this principle, the total mechanical energy of a system remains constant when only conservative forces (like gravity) are acting on it.
The total mechanical energy of an object can be calculated by adding its potential energy and kinetic energy. In this case, Lauren's potential energy at the bottom of the steep hill will be converted entirely to kinetic energy at the top of the next hill.
The potential energy (PE) of an object can be calculated using the formula:
PE = mgh
Where:
m = mass of the object
g = acceleration due to gravity (approximately 9.8 m/s²)
h = height
Given that the height of the hill is 25 meters, we can now calculate the potential energy at the bottom of the hill using the equation above.
PE = mgh
PE = (mass of Lauren)(9.8 m/s²)(25 m)
Now we need to find Lauren's speed at the bottom of the hill. The kinetic energy (KE) of an object can be calculated using the formula:
KE = (1/2)mv²
Where:
m = mass of the object
v = speed
Since we want the speed at the bottom of the hill, where Lauren has no pedaling input, her initial kinetic energy will be zero. Therefore, the potential energy at the bottom of the hill should be equal to the kinetic energy at the top of the next hill.
PE = KE
(mass of Lauren)(9.8 m/s²)(25 m) = (1/2)(mass of Lauren)v²
Now let's simplify the equation by canceling out the mass of Lauren:
9.8 m/s²(25 m) = (1/2)v²
To solve for v, first calculate the right-hand side of the equation:
(1/2)v² = (9.8 m/s²)(25 m) × 2
v² = (9.8 m/s²)(25 m) × 2
v² = 490 m²/s² × 2
v² = 980 m²/s²
Finally, take the square root of both sides to solve for the speed:
v = √(980 m²/s²)
v ≈ 31.3 m/s
Therefore, Lauren's speed at the bottom of the hill needs to be approximately 31.3 m/s.