A ski jumper launches from a ski jump that is oriented parallel to a hill. The jump has a vertical drop of 50 m and the coefficient of friction μ between the skier and the jump is 0.05. The launch point is 5 m above the hill and there is a small lip at the bottom of the jump so that the skier launches horizontally. How long in seconds is the skier in flight?
Details and assumptions
Neglect air resistance.
The acceleration of gravity is −9.8 m/s2.
The hill slopes down at an angle of θ=30o below the horizontal.
The skier started from rest at the top of the jump.
3.795 seconds
To find the time the skier is in flight, we need to calculate the time it takes for the skier to fall vertically and horizontally.
First, let's find the time it takes for the skier to fall vertically. We can use the equation of motion:
Δy = v₀t + (1/2)gt²
where:
Δy - vertical displacement (-50 m)
v₀ - initial vertical velocity (0 m/s, since the skier starts from rest)
g - acceleration due to gravity (-9.8 m/s²)
t - time
Rearranging the equation, we have:
t = √(-2Δy/g)
Plugging in the values, we get:
t = √(-2(-50)/-9.8)
t = √(100/9.8)
t ≈ 32.30 seconds
Next, let's find the horizontal distance the skier travels during the flight. We can use the equation:
Δx = v₀x * t
where:
Δx - horizontal displacement
v₀x - initial horizontal velocity (since the skier launches horizontally, v₀x = 0 m/s)
t - time
Since the skier starts with zero horizontal velocity, the horizontal displacement is related to the horizontal component of the initial velocity:
Δx = v₀cos(θ) * t
where θ is the angle of the hill (30°).
We know that the skier starts 5 m above the hill, so the total horizontal displacement is the distance along the hill plus an additional 5 m:
Δx_total = Δx + 5 m
Using the equation, we can find Δx:
Δx = v₀cos(θ) * t
Δx = 0 * cos(30°) * t
Δx = 0
Therefore, Δx_total = 0 + 5 m = 5 m.
Since the skier does not travel horizontally during the flight, the skier is in the air for 0 seconds horizontally.
Therefore, the skier is in flight for approximately 32.30 seconds vertically and 0 seconds horizontally.
Time depends only on gravity.
50=1/2 × gt^2 solve for time t
the skier starts horizontally