A skier is accelerating down a 30.0° hill at a = 4.10 m/s2

a) What is the vertical component of her acceleration?

b) How long will it take her to reach the bottom of the hill, assuming she starts from rest and accelerates uniformly, if the elevation change is 340 m?

To find the answers to these questions, we can use the concepts of acceleration and motion along an inclined plane.

a) The vertical component of acceleration can be found by using the formula:

Acceleration (vertical) = Acceleration * sin(angle)

Given that acceleration is 4.10 m/s² and the angle is 30°, we can calculate the vertical component of acceleration:

Acceleration (vertical) = 4.10 m/s² * sin(30°)

To find the sin(30°), we can use a calculator or refer to a trigonometric table. The sin(30°) value is 0.5.

So, the vertical component of acceleration is:

Acceleration (vertical) = 4.10 m/s² * 0.5 = 2.05 m/s²

b) To find the time it takes for the skier to reach the bottom of the hill, we can use the equations of motion. Since the skier starts from rest, the initial velocity (u) is 0 m/s.

Using the equation:

Distance = (Initial Velocity * Time) + (0.5 * Acceleration * Time²)

We can rearrange the equation to solve for time:

Time = sqrt((2 * Distance) / Acceleration)

Given that the elevation change (distance) is 340 m and the acceleration is 4.10 m/s², we can substitute these values into the equation:

Time = sqrt((2 * 340 m) / 4.10 m/s²)

Calculating this equation gives us:

Time = sqrt(680 / 4.10) ≈ sqrt(166.34) ≈ 12.89 seconds

Therefore, it will take approximately 12.89 seconds for the skier to reach the bottom of the hill.

To solve this problem, we need to break down the acceleration into its components and then use the equations of motion.

Given:
Angle of the hill (θ) = 30.0°
Acceleration (a) = 4.10 m/s^2
Elevation change (h) = 340 m

a) To find the vertical component of acceleration, we can use the equation:

vertical acceleration (av) = a * sin(θ)

Substituting the given values:

vertical acceleration (av) = 4.10 m/s^2 * sin(30.0°)

Using a calculator:

av ≈ 2.05 m/s^2

Therefore, the vertical component of her acceleration is approximately 2.05 m/s^2.

b) To find the time it will take her to reach the bottom of the hill, we can use the equation:

h = (1/2) * av * t^2

Rearranging the equation to solve for time (t):

t^2 = (2 * h) / av
t = √((2 * 340 m) / 2.05 m/s^2)
t = √(680 m / 2.05 m/s^2)
t ≈ √(332.683 m²/s²)
t ≈ 18.24 s

Therefore, it will take her approximately 18.24 seconds to reach the bottom of the hill.