A skier of mass 109 kg travels down a frictionless ski trail.
a)If the top of the trail is a height 218 m above the bottom, what is the work done by gravity on the skier? _____J
b)Find the velocity of the skier when he reaches the bottom of the ski trail. Assume he starts from rest. ____m/s
To answer part a) of the question, we can use the concept of work done by gravity. The work done by gravity can be calculated using the formula:
Work = force x distance x cos(theta)
Where:
- force is the component of the force acting parallel to the direction of motion (in this case, the force of gravity),
- distance is the displacement along the direction of motion (in this case, the vertical distance),
- theta is the angle between the force vector and the displacement vector (in this case, 0 degrees or cos(0) = 1).
In this scenario, the force of gravity is given by the equation:
force of gravity = mass x acceleration due to gravity
The acceleration due to gravity is approximately 9.8 m/s^2.
Now, let's apply these values to calculate the work done by gravity:
Work = (mass x acceleration due to gravity) x distance x cos(0)
= (109 kg x 9.8 m/s^2) x 218 m x 1
Simplifying the equation, we get:
Work = 109 kg x 9.8 m/s^2 x 218 m
Evaluating this expression will give us the answer to part a) of the question.
To answer part b) of the question, we need to use the principle of conservation of energy. Initially, the skier is at the top of the ski trail with potential energy, but no kinetic energy. At the bottom of the ski trail, the skier will have no potential energy but will have gained kinetic energy.
The principle of conservation of energy states that the total mechanical energy of a system remains constant if no external forces are acting on it. In this case, we can assume that there are no external forces acting on the skier except gravity.
The total mechanical energy at the top of the ski trail is given by the equation:
Initial mechanical energy = Potential energy at the top
Potential energy = mass x acceleration due to gravity x height
Evaluating this expression will give us the initial mechanical energy.
The total mechanical energy at the bottom of the ski trail is given by the equation:
Final mechanical energy = Kinetic energy at the bottom
Kinetic energy = (1/2) x mass x velocity^2
Simplifying the equation, we get:
Final mechanical energy = (1/2) x mass x velocity^2
Since the total mechanical energy is conserved, we can equate the initial mechanical energy to the final mechanical energy:
Initial mechanical energy = Final mechanical energy
mass x acceleration due to gravity x height = (1/2) x mass x velocity^2
Simplifying the equation and rearranging it to solve for velocity, we get:
velocity = sqrt(2 x acceleration due to gravity x height)
Evaluating this expression will give us the answer to part b) of the question.