A 65-kg skier starts at rest and slides 30 m down a hill inclined at 12∘ relative to the horizontal. Consider the force of friction on the skier to be negligible.
Determine the speed of the skier after sliding 30 m downhill.
h = 30*sin12 = 6.24 m.
Vf^2 = Vo^2 + 2g*h = 0 + 19.6*6.24 = 122.3.
Vf = 11.1 m/s.
Well, it seems like this skier is going to have a downhill adventure! Let's calculate the speed.
First, let's break down the forces acting on the skier. Since the force of friction is negligible, the only force acting on the skier is the component of the gravitational force parallel to the hill.
The parallel component of the gravitational force can be calculated using the formula F = m * g * sinθ, where m is the mass of the skier, g is the acceleration due to gravity (approximately 9.8 m/s²), and θ is the angle of inclination (12∘).
So, the parallel component of the gravitational force is F = 65 kg * 9.8 m/s² * sin(12∘).
To find the velocity, we can use the equation v² = u² + 2aS, where v is the final velocity, u is the initial velocity (which is 0 m/s since the skier starts at rest), a is the acceleration, and S is the distance travelled (30 m in this case).
Since the skier is going downhill, the acceleration is the component of the gravitational force parallel to the hill divided by the mass of the skier (a = F/m):
a = (65 kg * 9.8 m/s² * sin(12∘)) / 65 kg.
Now we can plug the values into the equation:
v² = 0² + 2 * [(65 kg * 9.8 m/s² * sin(12∘)) / 65 kg] * 30 m.
Calculating the equation, we find the square of the velocity. To get the actual velocity, we'll just take the square root of that result.
But hey, before I give you a precise answer, let me just remind you that speed and velocity are very similar. The only difference is that velocity is a vector quantity and takes into account the direction. So, make sure you let us know whether you want the magnitude (speed) or the velocity (magnitude and direction).
In any case, after doing some calculations, the speed of the skier after sliding 30 m downhill is approximately... drumroll, please... 10.3 m/s. Enjoy the ride, skier!
To determine the speed of the skier after sliding down the hill, we can use the principles of basic kinematics and energy conservation.
Step 1: Determine the gravitational force acting on the skier.
The gravitational force acting on the skier can be calculated using the formula:
F_gravity = mass * gravity
where mass is the mass of the skier (65 kg) and gravity is the acceleration due to gravity (approximately 9.8 m/s²).
F_gravity = 65 kg * 9.8 m/s² = 637 N
Step 2: Determine the component of the gravitational force acting parallel to the incline.
The component of the gravitational force acting parallel to the incline can be calculated using the formula:
F_parallel = F_gravity * sin(θ)
where θ is the angle of the incline (12°).
F_parallel = 637 N * sin(12°) = 135.45 N
Step 3: Determine the work done by the parallel force.
The work done by the parallel force can be calculated using the formula:
Work = force * distance
where the force is the parallel component of the gravitational force (135.45 N) and the distance is the distance slid down the hill (30 m).
Work = 135.45 N * 30 m = 4063.5 J
Step 4: Determine the change in potential energy.
The change in potential energy can be calculated using the formula:
ΔPE = mass * gravity * height
where mass is the mass of the skier (65 kg), gravity is the acceleration due to gravity (9.8 m/s²), and height is the vertical distance slid down the hill. Since we are given that the hill is inclined at 12° relative to the horizontal, the height can be calculated as:
height = distance * sin(θ)
height = 30 m * sin(12°) ≈ 6.19 m
ΔPE = 65 kg * 9.8 m/s² * 6.19 m = 3831.54 J
Step 5: Apply the principle of energy conservation.
According to the principle of energy conservation, the work done by the parallel force (4063.5 J) is equal to the change in kinetic energy (ΔKE) plus the change in potential energy (ΔPE).
Work = ΔKE + ΔPE
4063.5 J = ΔKE + 3831.54 J
ΔKE = 4063.5 J - 3831.54 J = 231.96 J
Step 6: Determine the speed of the skier.
The change in kinetic energy (ΔKE) is equal to:
ΔKE = 1/2 * mass * velocity²
where mass is the mass of the skier (65 kg) and velocity is the final velocity of the skier.
231.96 J = 1/2 * 65 kg * velocity²
Simplifying the equation:
4639.2 J = 65 kg * velocity²
Dividing both sides by 65 kg:
71.37 m²/s² = velocity²
Taking the square root of both sides:
velocity ≈ √71.37 m²/s²
velocity ≈ 8.45 m/s
Therefore, the speed of the skier after sliding 30 m downhill is approximately 8.45 m/s.
To determine the speed of the skier after sliding 30 m downhill, we can use the principles of kinematics.
First, let's find the acceleration of the skier using the inclined plane. The acceleration of the skier down the hill is given by the equation:
a = g * sin(θ)
Where:
a is the acceleration
g is the acceleration due to gravity (approximately 9.8 m/s^2)
θ is the angle of the hill relative to the horizontal (12 degrees)
Plugging in the values, we get:
a = 9.8 m/s^2 * sin(12)
Next, we use the kinematic equation:
v^2 = u^2 + 2a * s
Where:
v is the final velocity (speed) of the skier
u is the initial velocity (speed) of the skier (which is 0, as the skier starts from rest)
a is the acceleration (which we calculated in the previous step)
s is the distance traveled (30 m)
Plugging in the values and solving for v, we get:
v^2 = 0^2 + 2 * (a from previous step) * 30
Now we can calculate the value of v.