A semi-trailer is coasting downhill along a mountain highway when its brakes fail. The driver pulls onto a runaway truck ramp that is inclined at an angle of 13.0° above the horizontal. The semi-trailer coasts to a stop after traveling 190 m along the ramp. What was the truck's initial speed? Neglect air resistance and friction.
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V^2 = Vo^2 + 2g.h
Vo^2 = V^2-2g.h
Vo^2 = 0-(-19.8)*190sin13 = 837.7
Vo = 28.9 m/s.
To find the initial speed of the truck, we can use the principles of physics and the given information.
First, let's break down the forces acting on the truck along the ramp. Since there is no friction or air resistance, the only force acting on the truck is its weight, which is directed straight down in the vertical direction. We can resolve this weight force into two components:
1. The component parallel to the incline, which contributes to the acceleration of the truck.
2. The component perpendicular to the incline, which is balanced by the normal force of the ramp.
Let's start by finding the component of the weight force parallel to the incline. We'll use the equation:
F_parallel = m * g * sin(θ)
where
F_parallel is the force parallel to the incline,
m is the mass of the truck, and
g is the acceleration due to gravity (approximately 9.8 m/s²).
Next, we can use Newton's second law of motion, which states that the net force acting on an object is equal to its mass multiplied by its acceleration. In this case, the net force acting on the truck is equal to the component of the weight force parallel to the incline. So we have:
F_net = F_parallel
Since the net force is equal to the mass of the truck multiplied by its acceleration, we can rewrite the equation as:
m * a = m * g * sin(θ)
The mass of the truck cancels out, leaving us with:
a = g * sin(θ)
Now let's find the acceleration of the truck. We know that the truck coasts to a stop after traveling a distance of 190 m along the ramp. When the truck comes to a stop, its final velocity is 0.
We can use the following equation of motion to find the acceleration:
v_f² = v_i² + 2 * a * d
where
v_f is the final velocity (0 m/s in this case),
v_i is the initial velocity, which is what we need to find,
a is the acceleration (from the previous equation),
and d is the distance traveled along the ramp (190 m).
Substituting the given values, the equation becomes:
0² = v_i² + 2 * (g * sin(θ)) * d
Simplifying further:
0 = v_i² + 2 * 9.8 m/s² * sin(13.0°) * 190 m
Now we can solve this equation for the initial velocity, v_i.
v_i² = - 2 * 9.8 m/s² * sin(13.0°) * 190 m
Taking the square root of both sides gives us:
v_i = √(- 2 * 9.8 m/s² * sin(13.0°) * 190 m)
Evaluating this expression using a calculator will give us the initial speed of the truck.