A truck of mass 9212 kg moving at a speed of 73.2 mph has lost its brakes. Fortunately, the driver finds a runaway lane, a gravel-covered incline that uses friction to stop a truck in such a situation; see the figure. In this case, the incline makes an angle of θ = 39.65° with the horizontal, and the gravel has a coefficient of friction of 0.634 with the tires of the truck. How far along the incline (∆x) does the truck travel before it stops?

To find the distance (∆x) that the truck travels before it stops, we need to break down the forces acting on the truck and use the principles of physics. Here's how you can solve this problem step-by-step:

Step 1: Convert the speed from mph to m/s:
Given the speed of the truck is 73.2 mph.
1 mph is approximately equal to 0.44704 m/s.
So, converting 73.2 mph to m/s:
73.2 mph * 0.44704 m/s = 32.705568 m/s (rounded to 6 decimal places).

Step 2: Resolve the forces acting on the truck:
The forces acting on the truck are:
- The gravitational force (mg), where m is the mass of the truck and g is the acceleration due to gravity (approximately 9.8 m/s^2).
- The normal force (N) exerted by the incline on the truck, perpendicular to the incline.
- The frictional force (f) between the tires of the truck and the gravel-covered incline, parallel to the incline.

Step 3: Calculate the gravitational force (mg):
The gravitational force acting on the truck is given by the formula:
Gravitational force (mg) = mass (m) * acceleration due to gravity (g).

Given the mass of the truck is 9212 kg and the acceleration due to gravity is 9.8 m/s^2:
Gravitational force = 9212 kg * 9.8 m/s^2 = 90373.6 N (rounded to 1 decimal place).

Step 4: Calculate the normal force (N):
The normal force exerted by the incline on the truck is equal to the component of the gravitational force perpendicular to the incline.
Normal force (N) = gravitational force (mg) * cos(θ).

Given the angle of the incline (θ) is 39.65°:
Normal force = 90373.6 N * cos(39.65°) = 90373.6 N * 0.7699 = 69596.59944 N (rounded to 5 decimal places).

Step 5: Calculate the frictional force (f):
The frictional force between the tires of the truck and the gravel-covered incline is given by:
Frictional force (f) = coefficient of friction (μ) * normal force (N).

Given the coefficient of friction (μ) is 0.634:
Frictional force = 0.634 * 69596.59944 N = 44108.36511 N (rounded to 5 decimal places).

Step 6: Calculate the net force (F_net):
The net force acting on the truck is given by:
Net force (F_net) = frictional force (f) - gravitational force (mg) * sin(θ).

Given the angle of the incline (θ) is 39.65°:
Net force = 44108.36511 N - 90373.6 N * sin(39.65°) = 44108.36511 N - 90373.6 N * 0.6336 = -15728.09073 N (rounded to 5 decimal places).

Note: The negative sign indicates that the net force is acting in the opposite direction of the motion.

Step 7: Calculate the acceleration (a):
The acceleration of the truck is given by Newton's second law of motion:
Net force (F_net) = mass (m) * acceleration (a).

Given the mass of the truck is 9212 kg:
Acceleration = Net force / mass = -15728.09073 N / 9212 kg = -1.706 m/s^2 (rounded to 3 decimal places).

Step 8: Calculate the stopping distance (∆x):
The stopping distance (∆x) can be determined using the equations of motion.

We can use the equation:
v_f^2 = v_i^2 + 2a∆x,

Where:
v_f is the final velocity (which is 0 m/s as the truck stops),
v_i is the initial velocity (which is 32.705568 m/s),
a is the acceleration (-1.706 m/s^2),
∆x is the stopping distance (the value we need to find).

Rearranging the equation to solve for ∆x:
∆x = (v_f^2 - v_i^2) / (2a).

Substituting the known values:
∆x = (0 m/s - (32.705568 m/s)^2) / (2 * -1.706 m/s^2) = 345.636465 m (rounded to 3 decimal places).

Step 9: Answer:
The truck will travel approximately 345.636 meters (rounded to 3 decimal places) along the incline before it comes to a stop.

To find the distance (∆x) the truck travels along the incline before it stops, we need to first determine the deceleration of the truck.

1. Convert the speed of the truck from miles per hour (mph) to meters per second (m/s):
- 1 mph = 0.44704 m/s
- Speed of the truck = 73.2 mph * 0.44704 m/s = 32.686368 m/s

2. Resolve the weight of the truck into components parallel and perpendicular to the incline:
- Perpendicular component (mg⊥) = mass (m) * acceleration due to gravity (g) * cos(θ)
- Parallel component (mg∥) = mass (m) * acceleration due to gravity (g) * sin(θ)
- Mass (m) = 9212 kg
- Acceleration due to gravity (g) = 9.8 m/s²
- Angle (θ) = 39.65°

Substitute the given values into the equations to find the perpendicular and parallel components of the weight of the truck.

3. Calculate the frictional force acting on the truck:
- Frictional force (Ff) = coefficient of friction (μ) * perpendicular component of weight (mg⊥)
- Coefficient of friction (μ) = 0.634 (given)

Substitute the calculated perpendicular component of the weight and the given coefficient of friction into the formula to find the frictional force.

4. Calculate the net force acting on the truck:
- Net force (Fnet) = frictional force (Ff) - parallel component of weight (mg∥)

Substitute the calculated frictional force and the calculated parallel component of the weight into the formula to find the net force.

5. Apply Newton's second law of motion to find the deceleration of the truck:
- Net force (Fnet) = mass (m) * deceleration (a)

Rearrange the equation to solve for the deceleration.

6. Use the kinematic equation to find the distance (∆x) the truck travels along the incline before it stops:
- Final velocity (v) = 0 m/s
- Initial velocity (u) = 32.686368 m/s
- Deceleration (a) = calculated deceleration from step 5

Substitute the given values into the kinematic equation: v² = u² + 2as
Rearrange the equation and solve for the distance (∆x) traveled.

By following these steps, you will be able to calculate the distance the truck travels along the incline before it stops.