An out-of-control truck with a mass of 5000kg is travelling at 35.0m/s (about 80mi/h) when it starts descending a steep (15 degrees), downward incline or slope. The incline is icy, so the coefficient of friction is 0.30. Use the work-energy theorem to determine how far the truck will skid. Assume it locks its brakes and skids the whole way before it comes to rest.
I'm not good with Work-Energy Theorem:(
N is given by mg cos (angle) = 5000 kg x 9.8 m/s^2 x cos (15) = 47330 N
So Ff = 47330 x 0.30 = 14,200 N energy of the truck is 1/2 (mv^2) = 3,062,500 J
14,200 x = 3,062,500
x=253.7 m
No worries, I can guide you step-by-step through the problem using the work-energy theorem.
The work-energy theorem states that the work done on an object is equal to the change in its kinetic energy. In this case, we'll use it to calculate the distance the truck will skid.
To solve this problem, we need to break it down into smaller steps:
Step 1: Determine the initial kinetic energy of the truck.
Step 2: Calculate the work done on the truck due to the friction force.
Step 3: Equate the work done to the change in kinetic energy.
Step 4: Solve for the distance the truck will skid.
Let's go step by step.
Step 1: Determine the initial kinetic energy of the truck.
The initial kinetic energy (KE_initial) of the truck can be calculated using the formula:
KE_initial = (1/2) * mass * velocity^2
Given:
mass (m) = 5000 kg
velocity (v) = 35.0 m/s
KE_initial = (1/2) * 5000 kg * (35.0 m/s)^2
Step 2: Calculate the work done on the truck due to the friction force.
The work done (W_friction) by the friction force can be calculated using the formula:
W_friction = force_friction * displacement
Since the truck skids the whole way before coming to rest, the displacement (s) is what we want to find.
The friction force (force_friction) can be calculated using the formula:
force_friction = coefficient_of_friction * normal_force
The normal force (normal_force) can be calculated using the formula:
normal_force = mass * gravitational_acceleration * cos(angle)
Given:
coefficient_of_friction (μ) = 0.30
angle (θ) = 15 degrees
First, we need to convert the angle from degrees to radians:
angle_radians = angle * (π / 180)
angle_radians = 15 degrees * (π / 180)
angle_radians ≈ 0.2618 radians
Normal force:
normal_force ≈ mass * gravitational_acceleration * cos(angle_radians)
normal_force ≈ 5000 kg * 9.8 m/s^2 * cos(0.2618)
Force of friction:
force_friction ≈ coefficient_of_friction * normal_force
force_friction ≈ 0.30 * (5000 kg * 9.8 m/s^2 * cos(0.2618))
Now, let's calculate the work done by the friction force:
W_friction = force_friction * displacement
Step 3: Equate the work done to the change in kinetic energy.
According to the work-energy theorem:
W_friction = ΔKE
Since the truck comes to rest, the final kinetic energy (KE_final) is zero:
ΔKE = KE_final - KE_initial
ΔKE = 0 - KE_initial
Step 4: Solve for the distance the truck will skid.
Now we can equate the work done by the friction force to the change in kinetic energy and solve for the displacement (s).
W_friction = ΔKE
(force_friction * displacement) = (0 - KE_initial)
Solve the equation for displacement (s):
displacement = (0 - KE_initial) / force_friction
Plug in the values we calculated for KE_initial and force_friction to solve for displacement:
displacement ≈ (0 - KE_initial) / force_friction
Once you have substituted the values, simplify the expression to find the displacement.
Please let me know if you need further assistance with the calculations.
No worries! I can help you understand and use the work-energy theorem to solve this problem step by step.
The work-energy theorem states that the work done on an object equals the change in its kinetic energy. In this case, the work is done by the frictional force as the truck skids downhill, and this work is equal to the change in the truck's kinetic energy.
To solve this problem, we need to calculate the initial kinetic energy of the truck and the work done by the frictional force.
1. Calculate the initial kinetic energy:
The formula for kinetic energy is: KE = 0.5 * mass * velocity^2
Substituting the given values, we have:
KE = 0.5 * 5000 kg * (35.0 m/s)^2
2. Calculate the work done by the frictional force:
The formula for work is: Work = force * distance * cos(theta)
In this case, the force is the frictional force, which can be calculated using the formula: frictional force = coefficient of friction * normal force
The normal force is equal to the weight of the truck, which can be calculated using the formula: weight = mass * gravity
Substituting the given values, we have:
Normal force = 5000 kg * 9.8 m/s^2
Frictional force = 0.3 * Normal force
Next, we need to calculate the distance over which the skidding occurs. Since the truck skids the whole way before coming to rest, this distance is what we want to find.
3. Substitute the known values into the work-energy theorem equation:
According to the work-energy theorem, the work done (by the frictional force) is equal to the change in kinetic energy:
Work = KE_initial - KE_final
Since the truck comes to rest, the final kinetic energy is zero. Therefore:
Work = KE_initial - 0
We can also express the work done by the frictional force as:
Work = Frictional force * Distance * cos(theta)
Now we can equate the two expressions for work and solve for the distance.
4. Solve for the distance:
Frictional force * Distance * cos(theta) = KE_initial
Substituting the known values, we have:
0.3 * Normal force * Distance * cos(theta) = KE_initial
Now we can substitute the formulas for normal force and initial kinetic energy and solve for distance.
I hope this explanation helps you understand how to use the work-energy theorem to solve the given problem. If you have any further questions or need clarification, please let me know!