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.
V^2 = Vo^2 + 2g.h
Vo^2 = V^2-2g.h
Vo^2 = 0-(-19.6)*190sin13 = 837.7
Vo = 28.9 m/s
To find the initial speed of the truck, we can use the principles of conservation of energy.
The energy of the truck at the top of the ramp would be converted into gravitational potential energy and kinetic energy when it reaches the bottom of the ramp.
Let's assume the mass of the truck is m and its initial speed is v.
The gravitational potential energy of the truck at the top of the ramp is given by the equation:
PE = mgh
Where m is the mass of the truck, g is the acceleration due to gravity (9.8 m/s²), and h is the vertical height of the ramp.
Since the ramp makes an angle of 13.0° with the horizontal, we can calculate the vertical height of the ramp using trigonometry:
h = length of the ramp * sin(angle)
Given that the truck travels 190 m along the ramp, we can calculate the vertical height:
h = 190 * sin(13.0°)
Next, we need to calculate the gravitational potential energy at the top of the ramp.
Next, the kinetic energy of the truck at the bottom of the ramp will be given by:
KE = 1/2 * m * v^2
According to the principle of conservation of energy, the gravitational potential energy must be equal to the kinetic energy at the bottom of the ramp:
PE = KE
mgh = 1/2 * m * v^2
Since the mass m is present on both sides of the equation, it cancels out. We can rearrange the equation to solve for v:
v^2 = 2gh
Take the square root of both sides to get:
v = √(2gh)
Now we can substitute the values we have to find the initial speed of the truck.
Plug in the values of g, h, and solve for v:
v = √(2 * 9.8 * h)
where h = 190 * sin(13.0°)
Finally, calculate the value of v to get the initial speed of the truck.
D = Vo^2*sin(2A)/g = 190 m.
Vo^2*sin(26)/9.8 = 190
Vo^2 = 190/sin26 = 433.4
Vo = 20.8 m/s.
CORRECTION:
Vo^2 = 9.8*190/sin26 = 4247.54
Vo = 65.2 m/s.