A novice skier, starting from rest, slides down a frictionless 24.0 incline whose vertical height is 215. How fast is she going when she reaches the bottom?
To find the speed of the skier when she reaches the bottom, we can make use of the principle of conservation of energy. Since there is no friction to dissipate energy, the total mechanical energy of the skier remains constant throughout the motion.
First, let's define the variables:
- m: mass of the skier (unknown)
- g: acceleration due to gravity (9.8 m/s^2)
- θ: angle of the incline (24.0° or 0.42 radians)
- h: vertical height of the incline (215 m)
- v: velocity/speed of the skier at the bottom (unknown)
The initial potential energy of the skier at the top of the hill is given by mgh, where m is the mass, g is the acceleration due to gravity, and h is the vertical height. The final kinetic energy of the skier at the bottom of the hill is given by (1/2)mv^2, where v is the velocity/speed.
Since there is no friction, the total mechanical energy at the top is equal to the total mechanical energy at the bottom. Therefore, we can equate the initial potential energy to the final kinetic energy:
mgh = (1/2)mv^2
Now, we can cancel the mass 'm' from both sides of the equation:
gh = (1/2)v^2
We can rearrange the equation to solve for v:
v^2 = 2gh
Finally, we take the square root of both sides to find the value of v:
v = √(2gh)
Substituting the given values into the equation:
v = √(2 * 9.8 m/s^2 * 215 m * sin(24.0°))
Evaluating this expression will give us the speed of the skier when she reaches the bottom.
What are the dimensions of your numbers? degrees? radians? centimeters? inches?
One of the first things you should have learned in physics is the importance of dimensions when describing physical properties.
Try using conservation of energy to solve the problem