A skier is gliding along at 7.49 m/s on horizontal, frictionless snow. He suddenly starts down a 14.4° incline. His speed at the bottom is 28.5 m/s. What is the length of the incline?
To find the length of the incline, we can use the equations of motion for linear motion along an inclined plane.
First, let's determine the initial velocity of the skier along the incline. Since the initial velocity is given as 7.49 m/s on a horizontal surface, and there is no friction, the initial velocity along the incline remains the same.
The initial velocity along the incline (u) = 7.49 m/s
Next, let's find the final velocity along the incline (v). The final velocity is given as 28.5 m/s.
The final velocity along the incline (v) = 28.5 m/s
Now, let's find the angle of the incline (θ) which is given as 14.4°.
The angle of the incline (θ) = 14.4°
Using the equation of motion for linear motion along an inclined plane:
v^2 = u^2 + 2as
where:
v = final velocity along the incline
u = initial velocity along the incline
a = acceleration along the incline (due to gravity)
s = length of the incline
We need to find the acceleration along the incline (a). Since the skier is moving downhill, gravity is acting in the same direction as the motion, so the acceleration along the incline is equal to the acceleration due to gravity (g).
The acceleration along the incline (a) = acceleration due to gravity (g)
Acceleration due to gravity (g) = 9.8 m/s^2
Now we can substitute the values into the equation and solve for s:
v^2 = u^2 + 2as
(28.5 m/s)^2 = (7.49 m/s)^2 + 2 × (9.8 m/s^2) × s
810.25 m^2/s^2 = 55.9601 m^2/s^2 + 19.6 m/s^2 × s
810.25 m^2/s^2 - 55.9601 m^2/s^2 = 19.6 m/s^2 × s
754.29 m^2/s^2 = 19.6 m/s^2 × s
s = (754.29 m^2/s^2)/(19.6 m/s^2)
s ≈ 38.52 m
Therefore, the length of the incline is approximately 38.52 meters.