Starting from rest, a 75 kg skier slides down a 17.0° slope. If the coefficient of kinetic friction between the skis

and snow is 0.120 and it takes 16.9 s to get to the bottom, how long is the ski trail? (Use g = 9.8 m/s2.)

To find the length of the ski trail, we can break down the skier's motion into two components:

1. Motion parallel to the slope (downhill)
2. Motion perpendicular to the slope (normal force)

Let's start by finding the acceleration of the skier parallel to the slope.

The force of gravity acting on the skier is given by:
F_gravity = m * g, where m is the mass of the skier (75 kg) and g is the acceleration due to gravity (9.8 m/s^2).

The component of the force of gravity parallel to the slope is:
F_parallel = F_gravity * sin(θ), where θ is the angle of the slope (17.0°).

The frictional force acting on the skier is given by:
F_friction = μ * F_n, where μ is the coefficient of kinetic friction (0.120) and F_n is the normal force.

The normal force is given by:
F_n = F_gravity * cos(θ)

Since the skier is moving down the slope, the acceleration parallel to the slope is given by:
a_parallel = (F_parallel - F_friction) / m

To find the time it takes for the skier to reach the bottom of the slope, we can use the kinematic equation:
s = ut + (1/2) * a * t^2

where s is the distance traveled, u is the initial velocity, a is the acceleration, and t is the time. Since the skier starts from rest, the initial velocity is 0. We can rearrange the equation to solve for time:
t = sqrt(2s / a)

Now, we can substitute the given values into the equations and solve for the unknown variables.

1. Calculate the acceleration parallel to the slope:
F_gravity = m * g
F_parallel = F_gravity * sin(θ)
F_n = F_gravity * cos(θ)
F_friction = μ * F_n
a_parallel = (F_parallel - F_friction) / m

2. Calculate the time to reach the bottom of the slope:
t = sqrt(2s / a_parallel)

3. Solve for the length of the ski trail (s):
s = a_parallel * t^2 / 2

Let's calculate the values step-by-step.

To find the length of the ski trail, we can use the following steps:

Step 1: Calculate the gravitational force acting on the skier.
The gravitational force (Fg) can be calculated using the formula:
Fg = m * g
where m is the mass of the skier (75 kg) and g is the acceleration due to gravity (9.8 m/s^2).

Fg = 75 kg * 9.8 m/s^2
Fg = 735 N

Step 2: Calculate the component of the gravitational force parallel to the slope.
The component of the gravitational force parallel to the slope can be calculated using the formula:
F_parallel = Fg * sin(theta)
where theta is the angle of the slope (17.0°).

F_parallel = 735 N * sin(17.0°)
F_parallel = 735 N * 0.290
F_parallel = 213.15 N (rounded to two decimal places)

Step 3: Calculate the force of kinetic friction.
The force of kinetic friction (F_friction) can be calculated using the formula:
F_friction = coefficient * F_normal
where the coefficient of kinetic friction (μ) is given as 0.120 and the normal force (F_normal) is equal to the perpendicular component of gravitational force, which can be calculated using the formula:
F_normal = Fg * cos(theta).

F_normal = 735 N * cos(17.0°)
F_normal = 735 N * 0.957
F_normal = 703.895 N (rounded to three decimal places)

F_friction = 0.120 * 703.895 N
F_friction = 84.47 N (rounded to two decimal places)

Step 4: Calculate the net force.
The net force (F_net) acting on the skier can be calculated using the formula:
F_net = F_parallel - F_friction.

F_net = 213.15 N - 84.47 N
F_net = 128.68 N (rounded to two decimal places)

Step 5: Calculate the acceleration.
The acceleration (a) can be calculated using the formula:
F_net = m * a.

128.68 N = 75 kg * a
a = 128.68 N / 75 kg
a = 1.716 m/s^2 (rounded to three decimal places)

Step 6: Calculate the final velocity.
The final velocity (vf) of the skier can be calculated using the formula:
vf = vi + a * t,
where vi is the initial velocity (which is 0 as the skier starts from rest) and t is the time taken to reach the bottom (16.9 s).

vf = 0 + 1.716 m/s^2 * 16.9 s
vf = 29.003 m/s (rounded to three decimal places)

Step 7: Calculate the distance traveled.
The distance traveled can be calculated using the formula:
distance = (vi + vf) / 2 * t,
where vi is the initial velocity (0 m/s), vf is the final velocity (29.003 m/s), and t is the time taken to reach the bottom (16.9 s).

distance = (0 m/s + 29.003 m/s) / 2 * 16.9 s
distance = 435.4215 meters (rounded to four decimal places)

Therefore, the ski trail is approximately 435.4 meters.

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