3. In the figure, a runaway truck with failed brakes is moving downgrade at 107 km/h just before the driver steers the truck up a frictionless emergency escape ramp with an inclination of θ = 14°. The truck's mass is 1.4 × 104 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.) What is the minimum length L if (b) the truck's mass is decreased
To solve this problem, we need to use Newton's second law of motion and some basic principles of physics.
(a) To find the minimum length L required for the truck to stop momentarily along the escape ramp, we need to consider the forces acting on the truck.
1. First, let's consider the forces acting on the truck before it enters the escape ramp:
- The weight of the truck acting vertically downwards, given by W = mg, where m is the mass of the truck (1.4 × 10^4 kg) and g is the acceleration due to gravity (9.8 m/s^2).
- The gravitational force component acting parallel to the incline, given by F_parallel = mg sin(θ), where θ is the inclination angle of the ramp (14°).
- The initial kinetic energy of the truck due to its velocity (107 km/h) before entering the ramp.
2. Next, let's consider the forces acting on the truck while it moves up the frictionless escape ramp:
- The normal force exerted by the ramp on the truck perpendicular to the incline, which balances the weight of the truck. Since the ramp is frictionless, there is no force of friction.
3. For the truck to momentarily stop, the net force acting on it must be zero. Therefore, we can set up the following equation:
F_parallel = mg sin(θ)
4. Rearrange the equation to solve for the minimum length L of the ramp:
L = F_parallel / (mg sin(θ))
Substitute the given values:
L = (mg sin(θ)) / (mg sin(θ))
L = 1 / sin(θ)
Substituting the given values of θ = 14°, we can calculate the length L required.
(b) To find the new minimum length L if the truck's mass is decreased, you would repeat the steps above using the new mass value.
Remember to convert units consistently throughout the calculations.