Sam, whose mass is 75 kg, straps on his skis and starts down a 63 m -high, 20° frictionless slope. A strong headwind exerts a horizontal force of 200 N on him as he skies. (Horizontal relative to level ground, not aligned with the slope.) What is his speed at the bottom?
remember to deduct the work done by the wind
63 m * 200 N * cos(20º)
KE = PE.
0.5M*V^2 = M*g*h,
Divide both sides by M:
0.5V^2 = g*h,
0.5V^2 = 9.8 * 63,
V = 35 m/s.
To find Sam's speed at the bottom of the slope, we need to use the principles of mechanical energy conservation.
Step 1: Calculate the potential energy at the top of the slope:
Potential energy (PE) = mass * gravity * height
PE = 75 kg * 9.8 m/s² * 63 m
Step 2: Calculate the work done by the headwind:
Work (W) = force * distance
W = 200 N * 63 m
Step 3: Calculate the kinetic energy at the bottom of the slope:
Kinetic energy (KE) = (1/2) * mass * velocity²
Since we want to find the velocity, we need to set the total mechanical energy at the top equal to the total mechanical energy at the bottom:
PE (top) + W (headwind) = KE (bottom)
Now, let's solve the equation:
PE (top) = KE (bottom) - W (headwind)
75 kg * 9.8 m/s² * 63 m = (1/2) * 75 kg * velocity² - (200 N * 63 m)
Now, simplify and solve for velocity:
44100 J = (1/2) * 75 kg * velocity² - 12600 J
56700 J = (1/2) * 75 kg * velocity²
75600 J = 75 kg * velocity²
75600 J / 75 kg = velocity²
1008 m²/s² = velocity²
Finally, take the square root of both sides to solve for the velocity:
velocity = √1008 m²/s²
velocity ≈ 31.8 m/s
Therefore, Sam's speed at the bottom of the slope is approximately 31.8 m/s.
To find Sam's speed at the bottom of the slope, we can use the law of conservation of energy. The total mechanical energy of the system is conserved, which means that the initial potential energy at the top of the slope will be converted into kinetic energy at the bottom.
First, we need to calculate the potential energy at the top of the slope. The potential energy (PE) is given by the equation:
PE = mgh
where:
m = mass of the object (75 kg)
g = acceleration due to gravity (9.8 m/s^2)
h = height of the slope (63 m)
PE = 75 kg * 9.8 m/s^2 * 63 m
PE = 45,585 J
Next, we need to calculate the horizontal distance traveled by Sam. The horizontal distance can be calculated using trigonometry, since we know the angle of the slope (20°) and the height of the slope (63 m).
Horizontal distance (d) = height (h) / sin(angle)
d = 63 m / sin(20°)
d ≈ 183.73 m
Now, we can calculate the work done by the headwind. The work done (W) is given by the equation:
W = force (F) * distance (d)
where:
F = force exerted by the headwind (200 N)
d = horizontal distance traveled by Sam (183.73 m)
W = 200 N * 183.73 m
W = 36,746 J
Since the frictionless slope does not exert any work, the total work done is equal to the change in potential energy:
W + PE = ΔPE
ΔPE = 36,746 J + 45,585 J
ΔPE = 82,331 J
Since the potential energy is converted to kinetic energy at the bottom of the slope, we can calculate the kinetic energy (KE) using the equation:
KE = 0.5 * mass * velocity^2
We know that the initial kinetic energy is zero (as Sam starts from rest), so the final kinetic energy is equal to the change in potential energy.
KE = ΔPE
0.5 * mass * velocity^2 = 82,331 J
Solving for velocity:
velocity^2 = (2 * ΔPE) / mass
velocity^2 = (2 * 82,331 J) / 75 kg
velocity^2 ≈ 2198.14 m^2/s^2
velocity ≈ √(2198.14 m^2/s^2)
velocity ≈ 46.85 m/s
Therefore, Sam's speed at the bottom of the slope is approximately 46.85 m/s.