Truck brakes can fail if they get too hot. In some mountainous areas, ramps of loose gravel are constructed to stop runaway trucks that have lost their brakes. The combination of a slight upward slope and a large coefficient of friction in the gravel brings the truck safely to a halt.
Suppose a gravel ramp slopes upward at 4.70° and the coefficient of friction is 0.410. Find the length of a ramp that will stop a 14980.0 kg truck that enters the ramp at 37.0 m/s.
The gravel will work efficiently if the brakes of the truck are at least working, even though they may not be in perfect condition.
The kinetic energy, Ek of the truck on entering the ramp
=(1/2)mv²
Increase in potential energy, Ep, at the end of the ramp of length L
=mgLsin(θ)
Work done, W, against friction
=μmgcos(θ)*L
Equate energies and work done:
Ek=Ep+W
and solve for L.
I get L=142 m. approximately.
Check my work.
To find the length of the ramp, we can use the equation for the gravitational force acting on the truck along the ramp:
F_gravity = m * g * sin(theta)
where:
m = mass of the truck (14980.0 kg)
g = acceleration due to gravity (9.8 m/s^2)
theta = angle of the ramp (4.70°)
Next, we can find the frictional force acting on the truck along the ramp using the equation:
F_friction = m * g * cos(theta) * mu
where:
mu = coefficient of friction (0.410)
theta = angle of the ramp (4.70°)
Since we want to find the length of the ramp, we need to find the total force acting on the truck along the ramp. This force is the sum of the gravitational force and the frictional force:
F_total = F_gravity + F_friction
Once we have the total force, we can use the work-energy principle to find the work done on the truck as it comes to a stop:
Work = F_total * distance
Since the work done is equal to the initial kinetic energy of the truck, we can set up the equation:
(1/2) * m * v^2 = Work
where:
m = mass of the truck (14980.0 kg)
v = initial velocity of the truck (37.0 m/s)
To find the distance, we rearrange the equation and substitute the expressions for the total force and work:
distance = (1/2) * m * v^2 / F_total
Now we can plug in the given values to calculate the length of the ramp:
theta = 4.70° = 4.70 * (pi/180) radians
mu = 0.410
m = 14980.0 kg
g = 9.8 m/s^2
v = 37.0 m/s
First, let's calculate the gravitational force:
F_gravity = m * g * sin(theta) = 14980.0 kg * 9.8 m/s^2 * sin(4.70 * (pi/180)) = approximately 1091.39 N
Next, let's calculate the frictional force:
F_friction = m * g * cos(theta) * mu = 14980.0 kg * 9.8 m/s^2 * cos(4.70 * (pi/180)) * 0.410 = approximately 4486.39 N
Now, let's calculate the total force:
F_total = F_gravity + F_friction = 1091.39 N + 4486.39 N = approximately 5577.77 N
Finally, let's calculate the length of the ramp:
distance = (1/2) * m * v^2 / F_total = (1/2) * 14980.0 kg * (37.0 m/s)^2 / 5577.77 N = approximately 101.35 m
Therefore, the length of the ramp needed to stop the truck is approximately 101.35 meters.
To find the length of the ramp that will stop the truck, we can use the principle of work and energy. The work done by friction is equal to the change in kinetic energy of the truck.
The work done by friction can be calculated using the formula:
Work = Force x Distance
The force of friction can be calculated using the formula:
Force = Mass x Acceleration
The acceleration can be calculated using the formula:
Acceleration = Gravitational Acceleration x Sine(θ) - Coefficient of Friction x Gravitational Acceleration x Cosine(θ)
Where:
- Gravitational Acceleration is the acceleration due to gravity (approximately 9.8 m/s^2).
- θ is the angle of the slope in radians (4.70° converted to radians is 0.08208).
The distance can be calculated using the formula:
Distance = (Final Velocity^2 - Initial Velocity^2) / (2 x Acceleration)
Where:
- Final Velocity is the velocity of the truck at the end of the ramp (approximately 0 m/s).
- Initial Velocity is the velocity of the truck at the beginning of the ramp (37.0 m/s).
Let's calculate the length of the ramp:
Step 1: Convert the angle from degrees to radians.
θ = 4.70° x (π / 180) = 0.08208 radians
Step 2: Calculate the acceleration.
Acceleration = (9.8 m/s^2) x sin(0.08208) - (0.410) x (9.8 m/s^2) x cos(0.08208) ≈ 1.703 m/s^2
Step 3: Calculate the distance.
Distance = (0 m/s)^2 - (37.0 m/s)^2 / (2 x 1.703 m/s^2) ≈ -380.57 m/s^2
Note: The negative sign indicates that the truck's velocity is decreasing, so the squared value of the initial velocity is actually greater than the squared value of the final velocity.
Step 4: Take the positive value of the distance.
Distance = |-380.57 m/s^2| ≈ 380.57 m
Therefore, the length of the ramp that will stop the truck is approximately 380.57 meters.