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 19.0° above the horizontal. The semi-trailer coasts to a stop after traveling 170 m along the ramp. What was the truck's initial speed? Neglect air resistance and friction.

To solve this problem, we can use the principles of projectile motion and energy conservation. Here's how you can find the truck's initial speed:

1. Break down the forces acting on the truck: Since the truck is on a ramp, there are two main forces acting on it: the force due to gravity and the normal force (perpendicular to the ramp's surface).

2. Resolve the force of gravity: The force due to gravity can be resolved into two components: one parallel to the ramp (mg(sinθ)) and the other perpendicular to the ramp (mg(cosθ)). Here, θ represents the angle of inclination, which is given as 19.0°.

3. Determine the net force along the ramp: The net force along the ramp is equal to the component of gravity along the ramp, which is mg(sinθ). As there is no force opposing the motion in this direction, it will accelerate the truck.

4. Apply kinematic equations: Using the equation of motion for an object with constant acceleration, we can find the acceleration along the ramp (a) using the known distance (s) traveled and the initial velocity (u). The equation to use is:
s = ut + (1/2)at^2
Rearrange the equation to solve for acceleration:
a = 2s / t^2

5. Find the time taken to stop: Now, determine the time it takes for the truck to come to a stop while traveling 170 m along the ramp. As we know the distance traveled (s) and acceleration (a), we can use the following kinematic equation:
v^2 = u^2 + 2as
Since the truck stops, the final velocity (v) is 0.
Therefore, 0 = u^2 + 2as
This equation can be rearranged to solve for time (t):
t = sqrt(2s / a)

6. Calculate the initial velocity: Finally, we can calculate the initial velocity (u) using the equation of motion:
u = at

Now, plug in the values given in the question, and calculate the truck's initial speed.