Determine the stopping distance for a skier moving down a slope with friction with an initial speed of 20.6m/s. Assume μk=0.178 and θ=5.27.
To determine the stopping distance for a skier moving down a slope with friction, we can use the equations of motion and the concept of work and energy. The stopping distance can be calculated in two steps: first, calculating the distance traveled on the sloping surface, and secondly, calculating the distance traveled on a flat surface.
Step 1: Calculate the distance traveled on the sloping surface.
The equation to calculate the distance traveled on the sloping surface is given by:
d = (v_i^2 - v_f^2) / (2 * a),
where
d is the distance traveled,
v_i is the initial velocity,
v_f is the final velocity,
and a is the acceleration.
In this case, the initial velocity of the skier is 20.6 m/s, and the final velocity is 0 m/s since the skier is coming to a stop. The acceleration can be calculated using the equation:
a = g * sin(θ) - μk * g * cos(θ),
where
g is the acceleration due to gravity (approximately 9.8 m/s^2),
θ is the angle of the slope,
and μk is the coefficient of kinetic friction.
In this case, θ = 5.27 degrees, but we need to convert it to radians:
θ = 5.27 * π / 180.
Plugging in the values, we can calculate the acceleration:
a = 9.8 * sin(5.27 * π / 180) - 0.178 * 9.8 * cos(5.27 * π / 180).
Once we have the acceleration, we can calculate the distance traveled on the sloping surface using the equation mentioned earlier.
Step 2: Calculate the distance traveled on the flat surface.
To calculate the distance traveled on the flat surface, we need to determine the time taken to come to a stop on the sloping surface. We can use the equation:
v_f = v_i + a * t,
where
t is the time taken to come to a stop.
In this case, v_i is 0 m/s (since the skier is already at rest) and v_f is also 0 m/s (since the skier is coming to a stop).
Solving for t, we get:
t = -v_i / a.
Once we have the time taken to come to a stop, we can calculate the distance traveled on the flat surface using the equation:
d_flat = v_f * t.
Finally, the total stopping distance is the sum of the distance traveled on the sloping surface and the distance traveled on the flat surface.
Please plug in the values in the provided formulas and calculate the stopping distance for the given scenario.