Question

How to find the length of a driveway given mass, angle of slope, frictional force and final velocity

Answer 1

To find the length of a driveway given the mass of an object, the angle of slope, frictional force, and final velocity, you can use the concepts of energy and kinematics. Here is a method to do this:

### Step 1: Identify the Forces
1. **Weight of the object (W)**: \( W = mg \), where \( m \) is the mass and \( g \) is the acceleration due to gravity (\( \approx 9.81 \, \text{m/s}^2 \)).
2. **Component of weight acting down the slope**: \( W_{\text{parallel}} = mg \sin(\theta) \), where \( \theta \) is the angle of the slope.
3. **Frictional force**: It acts up the slope and is given as \( F_{\text{friction}} \).

### Step 2: Calculate the Net Force
The net force acting on the object along the slope can be found by subtracting the frictional force from the parallel component of weight:
\[
F_{\text{net}} = W_{\text{parallel}} - F_{\text{friction}} = mg \sin(\theta) - F_{\text{friction}}
\]

### Step 3: Apply Newton's Second Law
Using Newton's second law:
\[
F_{\text{net}} = ma
\]
where \( a \) is the acceleration of the object. Rearranging gives:
\[
a = \frac{F_{\text{net}}}{m} = \frac{mg \sin(\theta) - F_{\text{friction}}}{m} = g \sin(\theta) - \frac{F_{\text{friction}}}{m}
\]

### Step 4: Use Kinematic Equation
Assuming the object starts from rest, use the kinematic equation that relates acceleration, final velocity, and distance:
\[
v^2 = u^2 + 2a d
\]
where
- \( v \) is the final velocity,
- \( u \) is the initial velocity (0 if starting from rest),
- \( a \) is the acceleration,
- \( d \) is the distance (length of the driveway).

Substituting \( u = 0 \) simplifies this to:
\[
v^2 = 2ad
\]

Rearranging gives:
\[
d = \frac{v^2}{2a}
\]

### Step 5: Determine the Length of the Driveway
Now substitute the expression for \( a \):
\[
d = \frac{v^2}{2 \left(g \sin(\theta) - \frac{F_{\text{friction}}}{m}\right)}
\]

### Summary
So, the final expression for the length of the driveway is:
\[
d = \frac{v^2}{2 \left(g \sin(\theta) - \frac{F_{\text{friction}}}{m}\right)}
\]

You can plug in the values for \( v \), \( m \), \( g \), \( \theta \), and \( F_{\text{friction}} \) to calculate \( d \).