a truck with speed v1 = 14.0 m/s reaches the top of a mountain pass. It starts d

Question

a truck with speed v1 = 14.0 m/s reaches the top of a mountain pass. It starts down the other side, which is inclined ? = 4.50O above the horizontal. The truck?s brakes have failed, so as it travels a distance D = 3000 m the only force acting against its motion is a constant force of air resistance equal to 5.00% of the weight of the truck. The truck enters a ?runaway truck ramp?, an uphill incline at the side of the roadway designed to stop a truck in this situation. The ramp is made of gravel, which provides a coefficient of friction of 0.400. The truck travels L = 160 m along the ramp before coming to rest. Find ?, the angle the ramp makes with the horizontal.

Explanation / Answer

First, we find the velocity of the truck at the bottom of the hill:
The forces acting on it are gravity and air resistance.
The force from gravity in the down-hill direction is:

mgsin

The force from air resistance in the opposite direction is:

0.05m

So the resultant force is:

m(gsin - 0.05)

Since F = mA, that means the acceleration of the truck is:

gsin - 0.05

Now, we know that:

Vf^2 = Vi^2 + 2ad

So we can plug in the values that we know:

Vf^2 = 14^2 + 2*(9.8)*(sin(4.5 deg)-0.05)*(3000) = 1869.39

Vf = sqrt(1869.39) = 43.24 m/s

Now, we know that in the 160 m that it traveled up the ramp, it reached a final velocity of 0. Let's figure out what its acceleration must have been:

Vf^2 = Vi^2 + 2ad

(0)^2 = (43.24)^2 + 2*a*160

a = -(43.24)^2 / 320 = -5.843 m/s^2

Now we know the forces acting on it while it traveled up the ramp were gravity, air resitance and friction:

The force from gravity was:

mgsin in the negative direction

The air resistence was:

0.05m in the negative direction

The force from friction was:

0.400mgcos in the negative direction

So F = -mgsin - 0.05m - 0.400mgcos

F = m(-gsin - 0.05 - 0.400gcos)

So, A = -gsin - 0.05 - 0.400gcos

Now we can solve for theta:

-gsin - 0.05 - 0.400gcos = -5.843

Knowing that theta is between 0 and 90, the only solution is:

11.49 degrees

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