A 50.0-kg skier starting from rest travels 200 m down a hill that has a 20.0° slope. When the skier reaches the bottom of the hill, her speed is 30.0 m/s.
(a) How much work is done by friction as the skier comes down the hill?
(b) What is the magnitude of the friction force if the skier travels directly down the hill?
Oh, downhill skiing! Let's calculate the amount of work done by friction and the magnitude of the friction force, shall we?
(a) To calculate the work done by friction, we need to find the change in kinetic energy of the skier. We'll start by calculating the initial kinetic energy (K_i) and the final kinetic energy (K_f).
Since the skier starts from rest, the initial kinetic energy (K_i) is zero. We can calculate the final kinetic energy (K_f) using the formula:
K_f = (1/2) * m * v_f^2
where m is the mass of the skier (50.0 kg) and v_f is the final speed of the skier (30.0 m/s).
Plugging in the values, we get:
K_f = (1/2) * 50.0 kg * (30.0 m/s)^2
Now, let's calculate K_f:
K_f = 22,500 Joules
Since the change in kinetic energy is equal to the work done by friction (W_friction), we can say:
W_friction = K_f - K_i
W_friction = 22,500 Joules - 0 Joules
Therefore, the work done by friction as the skier comes down the hill is 22,500 Joules.
(b) If the skier travels directly down the hill, the slope would be at 0 degrees. In this case, there would be no vertical component of the skier's weight acting on the hill. Therefore, the magnitude of the friction force would also be zero.
But remember, this is just a hypothetical scenario and not very likely in reality.
Keep skiing safe and have fun!
To determine the work done by friction and the magnitude of the friction force, we need to use the concepts of work, energy, and Newton's laws of motion.
(a) The work done by friction can be calculated using the work-energy principle, which states that the work done on an object is equal to its change in kinetic energy.
The initial kinetic energy (KEi) is zero because the skier starts from rest. The final kinetic energy (KEf) can be calculated using the formula KEf = 0.5 * m * v^2, where m is the mass of the skier and v is the final velocity.
Given:
Mass of the skier (m) = 50.0 kg
Final velocity (v) = 30.0 m/s
KEf = 0.5 * 50.0 kg * (30.0 m/s)^2
KEf = 22500 J
Since the initial kinetic energy is zero, the work done by friction is equal to the skier's change in kinetic energy:
Work = KEf - KEi
Work = 22500 J - 0 J
Work = 22500 J
Therefore, the work done by friction as the skier comes down the hill is 22500 J.
(b) To find the magnitude of the friction force when the skier travels directly down the hill, we can use Newton's second law of motion. The force of friction can be calculated using the formula:
Force of friction (f) = m * a
Since the skier is traveling directly down the hill, the acceleration (a) can be found using the equation:
a = g * sin(theta)
Where g is the acceleration due to gravity (9.8 m/s^2) and theta is the slope angle (20 degrees).
a = 9.8 m/s^2 * sin(20 degrees)
a ≈ 3.34 m/s^2
Now, we can find the magnitude of the friction force:
f = 50.0 kg * 3.34 m/s^2
f ≈ 167 N
Therefore, the magnitude of the friction force when the skier travels directly down the hill is approximately 167 N.
To determine the work done by friction and the magnitude of the friction force, we can use the principles of work and energy.
(a) The work done by friction can be calculated using the work-energy principle. The work done by friction is equal to the change in the skier's kinetic energy.
First, let's calculate the initial kinetic energy (KE_i):
KE_i = 1/2 * m * v_i^2
where m = skier's mass = 50.0 kg
v_i = initial speed = 0 m/s (starting from rest)
KE_i = 1/2 * 50.0 kg * (0 m/s)^2
= 0 J
Next, let's calculate the final kinetic energy (KE_f):
KE_f = 1/2 * m * v_f^2
where v_f = final speed = 30.0 m/s
KE_f = 1/2 * 50.0 kg * (30.0 m/s)^2
= 22500 J
The change in kinetic energy (ΔKE) is given by:
ΔKE = KE_f - KE_i
= 22500 J - 0 J
= 22500 J
Since work done by friction is equal to the change in kinetic energy, the work done by friction is 22500 J.
(b) The magnitude of the friction force can be determined using the equation:
Friction force (F_friction) = μ * N
where μ = coefficient of friction and N = normal force.
In this case, since the skier is traveling directly down the hill, the normal force is equal to the skier's weight (mg), where g is the acceleration due to gravity.
N = m * g
= 50.0 kg * 9.8 m/s^2
= 490 N
The coefficient of friction (μ) is not provided in the question. To determine the magnitude of the friction force, we need the coefficient of friction.