A truck of mass 8912 kg moving at a speed of 66.7 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 = 36.75° 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 the truck travels before it stops, we need to calculate the gravitational force acting on the truck, the frictional force opposing its motion, and then use these forces to calculate the distance using the work-energy principle.
1. Calculate the gravitational force:
The gravitational force acting on the truck can be calculated using the formula Fg = m * g, where m is the mass of the truck and g is the acceleration due to gravity (9.8 m/s²).
Fg = 8912 kg * 9.8 m/s²
Fg = 87,481.6 N
2. Calculate the component of the gravitational force parallel to the incline:
Since the incline makes an angle of θ = 36.75° with the horizontal, the component of the gravitational force acting parallel to the incline is Fg_parallel = Fg * sin(θ).
Fg_parallel = 87,481.6 N * sin(36.75°)
Fg_parallel = 51,096.9 N
3. Calculate the frictional force:
The frictional force opposing the motion of the truck can be calculated using the formula F_friction = μ * F_normal, where μ is the coefficient of friction and F_normal is the normal force acting on the truck.
The normal force acting on the truck can be calculated using the formula F_normal = m * g * cos(θ).
F_normal = 8912 kg * 9.8 m/s² * cos(36.75°)
F_normal = 72,407.3 N
F_friction = 0.634 * 72,407.3 N
F_friction = 45,994.4 N
4. Calculate the net force acting on the truck:
The net force acting on the truck can be calculated using the formula F_net = F_parallel - F_friction.
F_net = Fg_parallel - F_friction
F_net = 51,096.9 N - 45,994.4 N
F_net = 5,102.5 N
5. Calculate the distance using the work-energy principle:
The work done by the net force is equal to the change in kinetic energy of the truck. The work done can be calculated using the formula W = F_net * ∆x, where ∆x is the distance traveled by the truck.
The change in kinetic energy can be calculated using the formula ∆K = (1/2) * m * v^2, where v is the initial velocity of the truck (66.7 mph converted to m/s).
∆K = (1/2) * 8912 kg * (66.7 mph * (1 m/s / 2.237 mph))^2
∆K = 0.5 * 8912 kg * (29.89 m/s)^2
∆K = 3,982,749.6 J
Since the gravel-covered incline is designed to completely stop the truck, the change in kinetic energy is equal to zero. Therefore, the work done by the net force is also zero.
W = 0, therefore, F_net * ∆x = 0
5,102.5 N * ∆x = 0
∆x = 0
Therefore, the truck stops immediately and does not travel any distance along the incline.
To determine the distance (∆x) the truck travels before it stops, we can use the concept of work and energy.
Here's how we can solve the problem step by step:
Step 1: Convert the speed from mph to m/s
Since the given speed is in mph, we need to convert it to meters per second (m/s) to work with the SI units. We can use the conversion: 1 mph = 0.44704 m/s.
So, the speed of the truck is:
v = 66.7 mph * 0.44704 m/s per mph = 29.7856488 m/s (rounded to 3 decimal places)
Step 2: Find the horizontal component of the truck's velocity
The horizontal component of the truck's velocity will determine the force opposing the truck's motion due to friction. In this case, the incline angle (θ) is given as 36.75°.
The horizontal component of the velocity (v_x) can be found using v_x = v * cos(θ).
Substituting the values:
v_x = 29.7856488 m/s * cos(36.75°) = 23.3140543 m/s (rounded to 3 decimal places)
Step 3: Calculate the gravitational force acting on the truck
The gravitational force (Fg) acting on the truck is given by the formula Fg = m * g, where m is the mass of the truck and g is the acceleration due to gravity (approximately 9.8 m/s²).
Substituting the values:
Fg = 8912 kg * 9.8 m/s² = 87393.6 N (rounded to 1 decimal place)
Step 4: Determine the opposing force due to friction
The opposing force (Ff) due to friction can be found using the formula Ff = μ * Fg, where μ is the coefficient of friction between the gravel and the truck's tires.
Substituting the values:
Ff = 0.634 * 87393.6 N = 55487.2224 N (rounded to 4 decimal places)
Step 5: Calculate the work done by friction
The work (W) done by friction can be found using the formula W = Fd, where F is the force and d is the distance traveled.
In this case, the force (F) opposing the truck's motion is Ff, and we need to find the distance (d) traveled to stop.
Since work done by friction results in a loss of the truck's kinetic energy, we can equate W = ΔKE, where KE is the initial kinetic energy of the truck.
The initial kinetic energy (KEi) of the truck can be calculated using the formula KEi = 0.5 * m * v².
Substituting the values:
KEi = 0.5 * 8912 kg * (29.7856488 m/s)² = 1966993.0783 J (rounded to 4 decimal places)
The final kinetic energy (KEf) of the truck is zero since it comes to a stop.
Using the equation W = ΔKE, we have:
W = KEi - KEf
W = 1966993.0783 J - 0 J
W = 1966993.0783 J
Step 6: Determine the distance traveled to stop
Using the formula W = Fd, we can rewrite it as:
d = W / Ff
Substituting the values:
d = 1966993.0783 J / 55487.2224 N = 35.4167239 m (rounded to 3 decimal places)
Therefore, the truck travels approximately 35.417 meters (rounded to 3 decimal places) along the incline before it stops.