An 80.0 kg rollerblader is at rest at the top of a 100 m hill at a 7o incline. The coefficient of friction on the hill is 0.110. What is the roller blader's kinetic energy at the bottom of the hill?
To find the rollerblader's kinetic energy at the bottom of the hill, we need to calculate the gravitational potential energy lost by the rollerblader as they slide down the hill and then convert it into kinetic energy.
The gravitational potential energy lost is given by the equation:
ΔPE = m * g * Δh
where:
m = mass of the rollerblader = 80.0 kg
g = acceleration due to gravity = 9.8 m/s^2
Δh = change in height = height of the hill = 100 m
ΔPE = 80.0 kg * 9.8 m/s^2 * 100 m
ΔPE = 78400 Joules
Now, let's calculate the rollerblader's final kinetic energy at the bottom of the hill.
The total mechanical energy (TE) at the top of the hill is equal to the initial potential energy, which is given by:
TE = PE_initial = m * g * h_initial
where:
m = mass of the rollerblader = 80.0 kg
g = acceleration due to gravity = 9.8 m/s^2
h_initial = height of the hill = 100 m
PE_initial = 80.0 kg * 9.8 m/s^2 * 100 m
PE_initial = 78400 Joules
The final mechanical energy (FE) at the bottom of the hill will be equal to the sum of the final kinetic energy (KE) and the final potential energy (PE_final).
FE = KE + PE_final
At the bottom of the hill, all the initial potential energy has been converted into both kinetic energy and potential energy due to gravity. The equation for the final mechanical energy becomes:
FE = KE + PE_final = 0 + m * g * h_final
where:
m = mass of the rollerblader = 80.0 kg
g = acceleration due to gravity = 9.8 m/s^2
h_final = height at the bottom of the hill = 0 m (since the rollerblader is at the bottom)
Plugging in the values, we get:
FE = 0 + 80.0 kg * 9.8 m/s^2 * 0 m
FE = 0 Joules
Since energy is conserved, the total mechanical energy (TE) at the top of the hill is equal to the final mechanical energy (FE) at the bottom of the hill.
TE = FE
PE_initial = 0 + m * g * h_final
Solving for the final potential energy (PE_final), we get:
PE_final = PE_initial - m * g * h_final
PE_final = 78400 Joules - 0 Joules
PE_final = 78400 Joules
Now, we can calculate the final kinetic energy (KE) at the bottom of the hill by subtracting the final potential energy (PE_final) from the total mechanical energy (TE).
KE = TE - PE_final
KE = 78400 Joules - 78400 Joules
KE = 0 Joules
Therefore, the rollerblader's kinetic energy at the bottom of the hill is 0 Joules, meaning they have no kinetic energy at that point.
To find the rollerblader's kinetic energy at the bottom of the hill, we need to calculate the change in potential energy and then convert it to kinetic energy.
The change in potential energy can be calculated using the formula:
ΔPE = m * g * h * sin(θ)
Where:
ΔPE is the change in potential energy
m is the mass of the rollerblader (80.0 kg)
g is the acceleration due to gravity (9.8 m/s^2)
h is the height of the hill (100 m)
θ is the incline angle (7°)
Let's calculate the change in potential energy:
ΔPE = 80.0 kg * 9.8 m/s^2 * 100 m * sin(7°)
Next, we need to calculate the work done by friction. The work done by friction can be calculated using the formula:
Work = μ * m * g * d
Where:
μ is the coefficient of friction (0.110)
m is the mass of the rollerblader (80.0 kg)
g is the acceleration due to gravity (9.8 m/s^2)
d is the distance traveled (100 m)
Let's calculate the work done by friction:
Work = 0.110 * 80.0 kg * 9.8 m/s^2 * 100 m
Now, we can calculate the total work done on the rollerblader by adding the work done by friction to the change in potential energy:
Total work = ΔPE + Work
Finally, we can convert the work done to kinetic energy using the formula:
Kinetic Energy = Total work
Now, you can plug in the values and calculate the rollerblader's kinetic energy at the bottom of the hill.
I assume 100 meters is the height of the hill, not the distance down the slope.
loss of potential energy
= 80 * 9.81 * 100
normal force on slope = 80 * 9.81 *cos 70
friction force = 0.110 * normal force
= 0.110 * 80 * 9,81 * cos 70
distance moved down slope
= 100 /sin 70
so work done against friction =
0.110*80*9,81*cos 70*100/sin 70
ke = m g h - work done on friction
(1/2) 80 v^2 = 80 * 9.81 * 100 - 0.110*80*9,81*cos 70*100/sin 70
cancel 80. Solve for v