A bicyclist on an old bike (combined mass: 96 kg) is rolling down (no pedaling or braking) a hill of height 103 m. Over the course of the 278 meters of downhill road, she encounters a constant friction force of 323 Newton. If her speed at the top of the hill is 7 m/s, what is her speed at the bottom of the hill?
To solve this problem, we can use the principle of conservation of mechanical energy. The total mechanical energy of the system (bicyclist and the bike) remains constant throughout the motion.
The mechanical energy at the top of the hill consists of two components: potential energy (due to the height) and kinetic energy (due to the speed). At the bottom of the hill, all of the potential energy is converted into kinetic energy.
The equation for conservation of mechanical energy is:
Potential Energy + Kinetic Energy = Potential Energy + Kinetic Energy
At the top of the hill:
Potential Energy(top) + Kinetic Energy(top) = Kinetic Energy(bottom)
The potential energy is given by the formula:
Potential Energy = mass * gravity * height
where mass is the combined mass of the bicycle and the rider, gravity is the acceleration due to gravity (9.8 m/s²), and height is the height of the hill (103 m).
The kinetic energy is given by the formula:
Kinetic Energy = (1/2) * mass * velocity²
where mass is the combined mass of the bicycle and the rider, and velocity is the speed of the bicyclist.
Let's first calculate the potential energy at the top of the hill:
Potential Energy(top) = mass * gravity * height
= 96 kg * 9.8 m/s² * 103 m
Next, calculate the kinetic energy at the top of the hill:
Kinetic Energy(top) = (1/2) * mass * velocity²
= (1/2) * 96 kg * (7 m/s)²
Now, we can use the conservation of mechanical energy equation to find the speed at the bottom of the hill:
Potential Energy(top) + Kinetic Energy(top) = Kinetic Energy(bottom)
Substituting the values we calculated, we have:
(96 kg * 9.8 m/s² * 103 m) + ((1/2) * 96 kg * (7 m/s)²) = (1/2) * 96 kg * velocity²
Simplifying the equation gives:
99792 + 2352 = 48 * velocity²
Solving for velocity gives:
velocity² = (99792 + 2352) / 48
velocity² = 2079
Taking the square root of both sides gives:
velocity = √2079
Therefore, the speed of the bicyclist at the bottom of the hill is approximately 45.6 m/s.