A skier is gliding along at 5.0{\rm m/s} on horizontal, frictionless snow. He suddenly starts down a 15{\rm ^{\circ}} incline. His speed at the bottom is 15{\rm m/s} . What is the length of the incline? How long does it take him to reach the bottom?
ΔKE=ΔPE
m•v²/2 - m•v₀²/2= m•g•Δh= m•g•s•sinα
s= (v² - v₀²)/2•g•sinα
s=at²/2 = ……
a=2s/t²
v=v₀ +at = v₀ + 2•s•t/t² = v₀ + 2•s/t
Solve for ‚t’
To find the length of the incline, we can use the concept of conservation of energy. When the skier is on the horizontal, frictionless snow, his initial kinetic energy is given by
KE1 = 0.5 * m * v1^2, where m is the mass of the skier and v1 is their initial speed.
When the skier is at the bottom of the incline, his final kinetic energy is given by
KE2 = 0.5 * m * v2^2, where v2 is their final speed.
Since energy is conserved, the initial kinetic energy is equal to the final kinetic energy.
KE1 = KE2
0.5 * m * v1^2 = 0.5 * m * v2^2
Canceling out the mass, we get
v1^2 = v2^2
Now we can solve for the length of the incline using trigonometry. The vertical component of the initial speed v1 is given by
v1_vertical = v1 * sin(15°)
The vertical component of the final speed v2 is given by
v2_vertical = v2 * sin(15°)
Since the vertical components of the initial and final speeds are equal (both are 0), we can set them equal to each other:
v1_vertical = v2_vertical
v1 * sin(15°) = v2 * sin(15°)
Canceling out sin(15°), we get
v1 = v2
Now we can solve for the length of the incline using the formula for distance:
distance = v1 * t
where t is the time taken to reach the bottom.
We're given that v1 = 5.0 m/s and v2 = 15 m/s. Plugging these values into the formula, we get
distance = 5.0 * t
To find the time taken, we use the formula of motion for uniformly accelerated motion:
v2 = v1 + a * t
where a is the acceleration. Since the incline is at an angle, the skier experiences an acceleration due to gravity acting along the incline:
a = g * sin(15°)
The acceleration due to gravity is approximately 9.81 m/s^2. Plugging in these values, we get
15 = 5.0 + (9.81 * sin(15°)) * t
Simplifying this equation, we find
(9.81 * sin(15°)) * t = 10
Solving for t, we get
t = 10 / (9.81 * sin(15°))
Finally, substituting the value of t back into the equation for distance, we get
distance = 5.0 * t
distance = 5.0 * (10 / (9.81 * sin(15°)))
Calculating this expression will give us the length of the incline.