Question
A 50 kg person is riding a toboggan down a 15.0 m high snowy hill. The coefficients of static and kinetic friction between the snow and the toboggan are 0.300 and 0.200, respectively. Determine the final speed of the person and toboggan at the bottom of the hill.
Answers
GPT-4o mini
To find the final speed of the person and toboggan at the bottom of the hill, we can use the principles of energy conservation with consideration of frictional forces acting against the motion.
### Step 1: Calculate Potential Energy at the Top
The potential energy (PE) at the top of the hill can be calculated using the formula:
\[
PE = mgh
\]
where \( m = 50 \) kg is the mass, \( g = 9.81 \) m/s² is the acceleration due to gravity, and \( h = 15.0 \) m is the height of the hill.
Calculating the potential energy:
\[
PE = 50 \, \text{kg} \times 9.81 \, \text{m/s}^2 \times 15.0 \, \text{m} = 7357.5 \, \text{J}
\]
### Step 2: Calculate the Forces Acting Against Motion
The friction force (f) opposing the motion can be calculated using the following formula:
\[
f = \mu_k N
\]
where \( \mu_k = 0.200 \) is the coefficient of kinetic friction and \( N \) is the normal force. The normal force on a flat surface is equal to the weight of the toboggan and rider:
\[
N = mg = 50 \, \text{kg} \times 9.81 \, \text{m/s}^2 = 490.5 \, \text{N}
\]
Calculating the friction force:
\[
f = 0.200 \times 490.5 \, \text{N} = 98.1 \, \text{N}
\]
### Step 3: Calculate Work Done by Friction
The work done by friction (W_f) as the toboggan goes down the hill can be calculated using the formula:
\[
W_f = f \cdot d
\]
where \( d = 15.0 \) m is the horizontal distance along the slope assuming steep slope equals height for simplicity.
Calculating work done by friction:
\[
W_f = 98.1 \, \text{N} \times 15.0 \, \text{m} = 1471.5 \, \text{J}
\]
### Step 4: Use Energy Conservation to Find Final Kinetic Energy
According to the conservation of energy, the initial potential energy minus the work done by friction will yield the final kinetic energy (KE) of the system:
\[
PE - W_f = KE
\]
Thus:
\[
7357.5 \, \text{J} - 1471.5 \, \text{J} = KE
\]
Calculating the kinetic energy:
\[
KE = 5886 \, \text{J}
\]
### Step 5: Find the Final Speed
The kinetic energy can also be expressed in terms of mass and velocity:
\[
KE = \frac{1}{2} mv^2
\]
Rearranging this to find the velocity \( v \):
\[
v = \sqrt{\frac{2 \cdot KE}{m}} = \sqrt{\frac{2 \cdot 5886 \, \text{J}}{50 \, \text{kg}}}
\]
Calculating the final speed:
\[
v = \sqrt{\frac{11772}{50}} = \sqrt{235.44} \approx 15.35 \, \text{m/s}
\]
### Final Result
Therefore, the final speed of the person and toboggan at the bottom of the hill is approximately:
\[
\boxed{15.35 \, \text{m/s}}
\]
### Step 1: Calculate Potential Energy at the Top
The potential energy (PE) at the top of the hill can be calculated using the formula:
\[
PE = mgh
\]
where \( m = 50 \) kg is the mass, \( g = 9.81 \) m/s² is the acceleration due to gravity, and \( h = 15.0 \) m is the height of the hill.
Calculating the potential energy:
\[
PE = 50 \, \text{kg} \times 9.81 \, \text{m/s}^2 \times 15.0 \, \text{m} = 7357.5 \, \text{J}
\]
### Step 2: Calculate the Forces Acting Against Motion
The friction force (f) opposing the motion can be calculated using the following formula:
\[
f = \mu_k N
\]
where \( \mu_k = 0.200 \) is the coefficient of kinetic friction and \( N \) is the normal force. The normal force on a flat surface is equal to the weight of the toboggan and rider:
\[
N = mg = 50 \, \text{kg} \times 9.81 \, \text{m/s}^2 = 490.5 \, \text{N}
\]
Calculating the friction force:
\[
f = 0.200 \times 490.5 \, \text{N} = 98.1 \, \text{N}
\]
### Step 3: Calculate Work Done by Friction
The work done by friction (W_f) as the toboggan goes down the hill can be calculated using the formula:
\[
W_f = f \cdot d
\]
where \( d = 15.0 \) m is the horizontal distance along the slope assuming steep slope equals height for simplicity.
Calculating work done by friction:
\[
W_f = 98.1 \, \text{N} \times 15.0 \, \text{m} = 1471.5 \, \text{J}
\]
### Step 4: Use Energy Conservation to Find Final Kinetic Energy
According to the conservation of energy, the initial potential energy minus the work done by friction will yield the final kinetic energy (KE) of the system:
\[
PE - W_f = KE
\]
Thus:
\[
7357.5 \, \text{J} - 1471.5 \, \text{J} = KE
\]
Calculating the kinetic energy:
\[
KE = 5886 \, \text{J}
\]
### Step 5: Find the Final Speed
The kinetic energy can also be expressed in terms of mass and velocity:
\[
KE = \frac{1}{2} mv^2
\]
Rearranging this to find the velocity \( v \):
\[
v = \sqrt{\frac{2 \cdot KE}{m}} = \sqrt{\frac{2 \cdot 5886 \, \text{J}}{50 \, \text{kg}}}
\]
Calculating the final speed:
\[
v = \sqrt{\frac{11772}{50}} = \sqrt{235.44} \approx 15.35 \, \text{m/s}
\]
### Final Result
Therefore, the final speed of the person and toboggan at the bottom of the hill is approximately:
\[
\boxed{15.35 \, \text{m/s}}
\]