A skier is gliding along at on horizontal, frictionless snow. He suddenly starts down a incline. His speed at the bottom is .
What is the length of the incline?
How long does it take him to reach the bottom?
His speed at the bottom is ?.
To find the length of the incline and the time taken to reach the bottom, we need to apply principles of physics, specifically kinematics and energy conservation.
Let's break down the problem into its components. We are given the initial speed of the skier (), the angle of the incline () and the speed of the skier at the bottom of the incline (). We need to find the length of the incline () and the time taken to reach the bottom ().
1. Length of the incline (): To find the length of the incline, we can use the concept of conservation of energy. The initial and final mechanical energy of the skier is the same since there is no external force doing work on the skier.
The initial mechanical energy (Ei) is given by:
Ei = (1/2)mv^2, where m is the mass of the skier and v is the initial speed
The final mechanical energy (Ef) is given by:
Ef = (1/2)mvf^2, where vf is the speed of the skier at the bottom
Since Ei = Ef, we can equate the two expressions and solve for ;
(1/2)mv^2 = (1/2)mvf^2
Given that v = and vf = , we can substitute these values into the equation and solve for :
(1/2)m()^2 = (1/2)m()^2
Simplifying the equation, we get:
()^2 = ()^2
Now, take the square root of both sides of the equation:
() = ()
Therefore, the length of the incline is ().
2. Time taken to reach the bottom (): To find the time taken to reach the bottom of the incline, we can use the equations of motion along the incline. Assuming there is no air resistance, the force acting down the incline is the gravitational force.
The gravitational force along the incline is given by:
F = mg sin()
This force is responsible for accelerating the skier down the incline. Using Newton's second law (F = ma), we can write:
mg sin() = ma
Since acceleration (a) is equal to the change in velocity (vf - vi) divided by the time taken (), we can rewrite the equation as:
mg sin() = m(vf - vi)/
Now, solving for :
= m(vf - vi)/(mg sin())
Substituting the given values, we get:
= ()/(g sin())
Therefore, the time taken to reach the bottom of the incline is ().
Please note that to obtain the final numerical values, you need to substitute the specific values given in the question for , , , , and g (acceleration due to gravity).