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 friction and 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.
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To find the time the skier is in flight, we need to determine the time it takes for the skier to reach the bottom of the jump and launch horizontally.
First, let's find the horizontal distance the skier travels from the launch point to the bottom of the jump.
The height of the jump can be divided into two components: the vertical drop of 50 m and the vertical distance of 5 m from the launch point above the hill.
The horizontal distance traveled can be determined using the formula:
distance = initial velocity * time + 0.5 * acceleration * time^2
Since the skier starts from rest at the top of the jump, the initial velocity is 0 m/s. The acceleration is the gravitational acceleration of -9.8 m/s^2.
For the vertical drop, the distance traveled is the vertical drop of 50 m, so:
50 = 0 * t + 0.5 * (-9.8) * t^2
Simplifying the equation, we get:
4.9 * t^2 = 50
Dividing both sides by 4.9, we get:
t^2 = 10
Taking the square root of both sides, we find:
t ≈ 3.16 seconds
Now let's calculate the time it takes for the skier to travel the 5 m distance from the launch point to the bottom of the jump.
Since the skier launches horizontally, there is no vertical displacement, only the horizontal distance needs to be considered.
The horizontal distance traveled can be determined using the formula:
distance = initial velocity * time
In this case, the distance is 5 m and the initial velocity is unknown. We need to find the initial velocity to calculate the time.
To find the initial velocity, we can use the coefficient of friction and the angle of the hill.
The frictional force can be calculated using the formula:
frictional force = coefficient of friction * normal force
The normal force can be calculated by decomposing the weight of the skier into vertical and horizontal components:
normal force = weight * cos(theta)
The weight of the skier can be calculated using the formula:
weight = mass * gravity
Since the mass of the skier is not given, we can assume a value of 70 kg.
Plugging in the values, we have:
weight = 70 kg * (-9.8 m/s^2) = -686 N (negative because it acts downwards)
normal force = (-686 N) * cos(30o) = -595.31 N (negative because it acts in the opposite direction of the weight)
frictional force = 0.05 * (-595.31 N) = -29.77 N (negative because it acts in the opposite direction of the motion)
The frictional force equals the force component that accelerates the skier horizontally:
frictional force = mass * acceleration
Since mass and acceleration are unknown, we set up the following equation:
-29.77 N = mass * acceleration
Now, let's find the acceleration using the equation:
acceleration = horizontal distance / time^2
Substituting the given values, we have:
-29.77 N = mass * (5 m / (3.16 s)^2)
Simplifying the equation, we get:
-29.77 N = mass * (5 m / 9.98 s^2)
Dividing both sides by (5 m / 9.98 s^2), we find:
acceleration = -29.77 N / (5 m / 9.98 s^2) ≈ -59.45 m/s^2
Now we can solve for the mass using the equation:
mass = acceleration / gravity
Substituting the values, we have:
mass = (-59.45 m/s^2) / (-9.8 m/s^2) ≈ 6.07 kg
With the mass known, we can now calculate the initial velocity:
-29.77 N = (6.07 kg) * velocity
Simplifying the equation, we get:
velocity ≈ -4.91 m/s (negative because the skier is moving in the opposite direction of the velocity)
Finally, we can calculate the time it takes for the skier to travel the 5 m distance:
5 m = (-4.91 m/s) * time
Simplifying the equation, we get:
time ≈ 1.02 seconds
Therefore, the total time the skier is in flight is approximately:
3.16 s + 1.02 s ≈ 4.18 seconds