Kathy is changing the tire of her car on a steep hill 20 m high. She trips
and drops the 10 kg spare tire which rolls down the hill. What is the speed of
the tire at the top of the next hill if the height of the hill is 5 m high?
To find the speed of the tire at the top of the next hill, we can use the principle of conservation of energy. The potential energy at the top of the first hill will be converted into kinetic energy at the top of the next hill.
First, let's calculate the potential energy of the tire at the top of the first hill:
Potential energy (PE) = mass (m) * acceleration due to gravity (g) * height (h)
PE = 10 kg * 9.8 m/s² * 20 m
PE = 1960 Joules
According to the conservation of energy, this potential energy will be converted into kinetic energy at the top of the next hill. The formula for kinetic energy is:
Kinetic energy (KE) = 0.5 * mass (m) * velocity² (v²)
We need to find the velocity (v). Rearranging the equation, we get:
v = √(2 * KE / m)
Now, let's calculate the kinetic energy at the top of the next hill:
KE = 0.5 * 10 kg * v²
Since the potential energy is completely converted to kinetic energy, we can equate the potential energy to the kinetic energy:
1960 Joules = 0.5 * 10 kg * v²
Now solve for v:
v² = (1960 Joules) / (0.5 * 10 kg)
v² = 392 m²/s²
v = √392 m/s
Therefore, the speed of the tire at the top of the next hill is approximately 19.8 m/s (rounded to one decimal place).
To solve this problem, we can use the conservation of mechanical energy. The potential energy lost by the tire as it descends the first hill is equal to the potential energy gained as it moves up the second hill. Let's calculate the speed of the tire at the top of the next hill:
1. Calculate the potential energy at the starting position (top of the first hill):
Potential Energy = mass * gravity * height
Potential Energy = 10 kg * 9.8 m/s² * 20 m
Potential Energy = 1960 Joules
2. Calculate the potential energy at the top of the second hill:
Potential Energy = mass * gravity * height
Potential Energy = 10 kg * 9.8 m/s² * 5 m
Potential Energy = 490 Joules
3. Since energy is conserved, the potential energy lost by the tire is equal to the potential energy gained:
Potential Energy lost = Potential Energy gained
1960 Joules = 490 Joules + Kinetic Energy at the top of the next hill
4. Rearrange the equation to solve for the kinetic energy:
Kinetic Energy at the top of the next hill = 1960 Joules - 490 Joules
Kinetic Energy at the top of the next hill = 1470 Joules
5. Convert the kinetic energy to speed:
Kinetic Energy = 0.5 * mass * velocity²
1470 Joules = 0.5 * 10 kg * velocity²
6. Solve for velocity:
velocity² = (2 * 1470 Joules) / 10 kg
velocity² = 2940 Joules / 10 kg
velocity² = 294 m²/s²
velocity = √(294 m²/s²)
velocity ≈ 17.14 m/s
Therefore, the speed of the tire at the top of the next hill is approximately 17.14 m/s.