A semitrailer 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 10.0° above the horizontal. The semitrailer coasts to a stop after traveling 137 m along the ramp. What was the truck's initial speed? Neglect air resistance and friction.
Answer will be in m/s.
m•v²/2=mgh=m•g•s•sinα
v=sqrt(2• g•s•sinα)
To find the truck's initial speed, we can use the principle of conservation of energy. Since no external forces (such as friction or air resistance) are acting on the truck, the total mechanical energy is conserved.
The initial mechanical energy of the truck can be calculated as the sum of its potential and kinetic energy:
E_initial = PE_initial + KE_initial
The potential energy is given by:
PE = m * g * h
Where m is the mass of the truck, g is the acceleration due to gravity (approximately 9.8 m/s²), and h is the vertical distance traveled along the inclined ramp.
The kinetic energy is given by:
KE = (1/2) * m * v²
Where m is the mass of the truck, and v is its initial velocity.
Since the truck comes to a stop at the end of the ramp, its final kinetic energy is zero. Therefore, the initial mechanical energy is equal to the potential energy:
E_initial = PE_initial = m * g * h
We can solve this equation for v:
(1/2) * m * v² = m * g * h
Canceling out the mass (m) on both sides:
(1/2) * v² = g * h
Simplifying further:
v² = 2 * g * h
Taking the square root of both sides:
v = √(2 * g * h)
Now we can substitute the given values into the equation:
g = 9.8 m/s²
h = 137 m
v = √(2 * 9.8 * 137)
≈ √2680.4
≈ 51.8 m/s
Therefore, the truck's initial speed was approximately 51.8 m/s.
To find the truck's initial speed, we can use the principles of projectile motion. The key here is to break down the motion into horizontal and vertical components.
Let's start by analyzing the motion along the ramp. We can assume the truck's initial velocity (v₀) is parallel to the ramp, and it will gradually decrease until it comes to a stop. We are given the distance traveled along the ramp (x = 137 m) and the angle of the ramp (θ = 10.0°).
To find the initial velocity, we need to determine the acceleration along the ramp. From the problem statement, we are told that there is no air resistance and friction, which means the only force acting on the truck along the ramp is its weight.
The weight of the truck can be represented by the force due to gravity (mg), where m is the mass of the truck and g is the acceleration due to gravity (approximately 9.8 m/s²). The weight can be resolved into two components: one parallel to the ramp (mg sin θ) and the other perpendicular to the ramp (mg cos θ). The component parallel to the ramp is responsible for the acceleration, while the perpendicular component is balanced by the normal force from the ramp.
The acceleration (a) along the ramp can be determined using Newton's second law, where the net force acting along the ramp is equal to the mass of the truck times its acceleration:
Net force along the ramp = mass of the truck × acceleration along the ramp
mg sin θ = m × a
Simplifying, we find that:
a = g sin θ
Now that we have the acceleration along the ramp, we can determine the initial velocity using the equations of motion.
The truck started from rest, so the initial velocity (v₀) is 0 m/s. The final velocity (v) is the velocity when the truck comes to a stop, and the distance traveled (x) is 137 m. We can use the equation:
v² = v₀² + 2a x
Since v₀ = 0, the equation simplifies to:
v² = 2ax
Substituting the known values for a and x, we get:
v² = 2(g sin θ) x
Finally, we can solve for v by taking the square root of both sides:
v = √(2(g sin θ) x)
Substituting the known values into the equation (g = 9.8 m/s², θ = 10.0°, x = 137 m), we can calculate the initial speed (v):
v = √(2(9.8 m/s²)(sin 10.0°)(137 m))
v ≈ 33.9 m/s
Therefore, the truck's initial speed was approximately 33.9 m/s.