In the figure below, a runaway truck with failed brakes is moving downgrade at 114 km/h just before the driver steers the truck travel up a frictionless emergency escape ramp with an inclination of 15°. The truck's mass is 5000 kg.
(a) What minimum length L must the ramp have if the truck is to stop (momentarily) along it? (Assume the truck is a particle, and justify that assumption.)
m
(b) Does the minimum length L increase, decrease, or remain the same if the truck's mass is decreased?
remain the same
decrease
increase
(c) Does the minimum length L increase, decrease, or remain the same if the truck's speed is decreased?
increase
decrease
remain the same
turn up bro
To solve this problem, we need to consider the forces acting on the truck and use the principles of Newtonian mechanics.
(a) To determine the minimum length L of the ramp, we need to find the maximum distance the truck can travel up the ramp before coming to a stop.
First, let's assume the truck is a particle. This assumption is valid if the dimensions of the truck are negligible compared to the length of the ramp. In this case, the truck's mass of 5000 kg is given, and we are assuming it behaves like a point mass.
Now, let's analyze the forces acting on the truck:
1. Weight (W): The weight of the truck acts vertically downward and can be calculated using the equation W = mg, where m is the mass of the truck (5000 kg) and g is the acceleration due to gravity (9.8 m/s^2).
W = (5000 kg) × (9.8 m/s^2) = 49,000 N
2. Normal force (N): The normal force acts perpendicular to the ramp and counterbalances the component of the weight that acts parallel to the ramp. Since the ramp is frictionless, the normal force is equal in magnitude but opposite in direction to the weight component parallel to the ramp.
N = mg × cos(θ), where θ is the angle of inclination (15°).
N = 5000 kg × 9.8 m/s^2 × cos(15°) ≈ 47,144 N
3. Net force (F_net): The net force acting on the truck is the difference between the gravitational force component parallel to the ramp and the normal force.
F_net = mg × sin(θ) - N
F_net = 5000 kg × 9.8 m/s^2 × sin(15°) - 47,144 N
F_net ≈ 11,894 N
According to Newton's second law (F_net = ma), where a is the acceleration of the truck, we can express the net force as:
F_net = ma
Now, if we assume that the truck comes to a momentary stop on the ramp, the acceleration of the truck will be zero. Therefore, we can set F_net equal to zero.
0 = ma
0 = m × 0
0 = 11,894 N
From this equation, we can see that the net force is equal to zero, indicating that the forces are balanced, and the truck momentarily stops on the ramp.
Now, let's determine the minimum length L of the ramp:
We know that the length of the ramp corresponds to the distance the truck will travel along the ramp before coming to a stop. We can use the equation for work to find this distance.
Work (W) is given by the formula:
W = force × distance × cos(θ)
In this case, the only force doing work is the component of the weight parallel to the ramp. So we can express the work done as:
W = mg × sin(θ) × d
Since the truck comes to a stop, the work done must be equal to the initial kinetic energy of the truck, which can be expressed as:
W = (1/2)mv^2, where v is the initial velocity of the truck (114 km/h).
Converting the velocity to m/s:
v = (114 km/h) × (1000 m/1 km) × (1 h/3600 s)
v ≈ 31.7 m/s
Setting the work done equal to the initial kinetic energy:
mg × sin(θ) × d = (1/2)mv^2
Simplifying the equation gives:
d = (v^2) / (2g × sin(θ))
d = (31.7 m/s)^2 / (2 × 9.8 m/s^2 × sin(15°))
d ≈ 231 m
So, the minimum length L of the ramp must be approximately 231 meters for the truck to stop momentarily along it.
(b) If the truck's mass is decreased, the minimum length L of the ramp will remain the same. This is because the mass does not affect the calculation for the minimum length of the ramp in this specific scenario.
(c) If the truck's speed is decreased, the minimum length L of the ramp will decrease. This is because the initial kinetic energy of the truck is directly proportional to the square of its velocity. As the speed decreases, the initial kinetic energy decreases, and thus, the minimum length of the ramp required to bring the truck to a stop will also decrease.