A 58 kg skier is coasting down a 250 slope. Near the top of the slope, her speed is 3.6 m/s. She accelerates down the slope because of the gravitational force, even though a kinetic frictional force of magnitude 71 N opposes her motion. Ignoring air resistance, determine the speed at a point that is displaced 57 m downhill.
-19m/s
To determine the speed at a point that is displaced 57 m downhill, we can use the principles of conservation of energy.
Here are the steps to calculate the speed:
Step 1: Calculate the gravitational potential energy at the initial point
The gravitational potential energy can be given by the formula:
PE = m * g * h
Where m is the mass of the skier (58 kg), g is the acceleration due to gravity (9.8 m/s²), and h is the vertical displacement (250 m).
Plugging in the values, we get:
PE = 58 kg * 9.8 m/s² * 250 m = 142,100 J
Step 2: Calculate the work done by the frictional force
The work done by the frictional force can be given by the formula:
W = -F * d
Where F is the force due to friction (71 N) and d is the displacement (250 m).
Plugging in the values, we get:
W = -71 N * 250 m = -17,750 J
Step 3: Calculate the final kinetic energy
Since energy is conserved, the sum of the initial potential energy and the work done by friction equals the final kinetic energy.
KE = PE + W
Plugging in the values, we get:
KE = 142,100 J + (-17,750 J) = 124,350 J
Step 4: Calculate the speed at the final point
The kinetic energy can also be given by the formula:
KE = 0.5 * m * v²
Where m is the mass of the skier (58 kg) and v is the final speed.
Rearranging the formula, we get:
v = sqrt(2 * KE / m)
Plugging in the values, we get:
v = sqrt(2 * 124,350 J / 58 kg) ≈ 11.6 m/s
Therefore, the speed at a point that is displaced 57 m downhill is approximately 11.6 m/s.
To determine the speed at a point that is displaced 57 m downhill, we can use the principles of motion along an inclined plane.
Let's break down the problem into steps:
Step 1: Calculate the net force acting on the skier
The net force acting on the skier is the vector sum of the gravitational force and the kinetic frictional force. Since the skier is moving downhill, the gravitational force and the kinetic frictional force are in the same direction.
Net Force = Force due to Gravity - Force of Kinetic Friction
Net Force = m * g - F_friction
where:
m = mass of the skier (58 kg)
g = acceleration due to gravity (9.8 m/s²)
F_friction = magnitude of the kinetic frictional force (71 N)
Substituting the values:
Net Force = 58 kg * 9.8 m/s² - 71 N
Net Force ≈ 547.6 N - 71 N
Net Force ≈ 476.6 N
Step 2: Calculate the acceleration of the skier
Using Newton's second law of motion, we can find the acceleration of the skier:
Net Force = mass * acceleration
476.6 N = 58 kg * acceleration
Rearranging the equation:
acceleration = 476.6 N / 58 kg
acceleration ≈ 8.21 m/s²
Step 3: Calculate the final velocity at a displaced point
We can now use the equation of motion for uniformly accelerated linear motion to find the final velocity at a displaced point.
v² = u² + 2as
where:
v = final velocity
u = initial velocity (3.6 m/s)
a = acceleration (-8.21 m/s²) since it acts in the opposite direction of the motion
s = displacement (57 m)
Substituting the values:
v² = (3.6 m/s)² + 2 * (-8.21 m/s²) * (57 m)
Calculating:
v² ≈ 12.96 m²/s² + (-939.54 m²/s²)
v² ≈ -926.58 m²/s²
Since the final velocity cannot be negative for this scenario, we discard the negative sign.
Taking the square root of both sides:
v ≈ √926.58 m²/s²
v ≈ 30.43 m/s
Therefore, the speed at a point that is displaced 57 m downhill is approximately 30.43 m/s.