91Ó°ÊÓ

An inclined plane is a flat surface tilted at an angle to the horizontal. It allows an object to move a longer distance with less effort than lifting it vertically. In the problem of the skier, the inclined plane is the snowy slope with an angle of 8 degrees. This setup is used in the exercise to demonstrate the effects of gravity acting on the skier. Inclined planes reduce the effort needed to move an object uphill but in this exercise, we look at the motion downhill. Here, gravity helps the skier accelerate from rest. Instead of moving directly downward, the skier descends a longer, gentler path because of the incline angle. When dealing with inclined planes:

• Gravity pulls objects directly downward, not along the slope.
• Inclined planes experience two components of gravitational force: one acts along the slope, moving the object downhill, and the other perpendicular to the slope.
• A key point to remember is: if the slope is frictionless, as in this problem, there is no resisting force.

Kinetic energy is the energy possessed by an object due to its motion. In simple terms, when an object moves, it has kinetic energy. The equation for kinetic energy is given by:\[ KE = \frac{1}{2}mv^2 \]where \( KE \) is kinetic energy, \( m \) is mass, and \( v \) is velocity. In the skier problem, as the skier moves down the frictionless incline, the potential energy at the top converts into kinetic energy. Initially, when the skier is at rest before sliding down, the kinetic energy is zero. But by the time she reaches the bottom, all the potential energy becomes kinetic energy, which makes her move faster.To determine how fast the skier is going at the bottom, we solve for velocity using the kinetic energy equation set equal to the initial potential energy, ensuring energy conservation:\[ mgh = \frac{1}{2}mv^2 \]Here, solving this equation helps us find that the kinetic energy at the bottom of the slope matches the potential energy at the top. This relationship is crucial in predicting how fast she is traveling when reaching the bottom.

Potential energy is the stored energy of an object due to its position or state. For objects at a height, like the skier at the top of the hill, this is gravitational potential energy. It is calculated using the formula:\[ PE = mgh \]where \( PE \) is potential energy, \( m \) is mass, \( g \) is the acceleration due to gravity (9.8 \( m/s^2 \)), and \( h \) is height. In the context of the skier problem, the skier starts at a height of 105 meters with significant potential energy due to her elevated position. This energy is what drives her to move as she descends the hill. As she goes down, her height decreases, converting potential energy into kinetic energy.Important points about potential energy:

• It’s directly proportional to both mass and height.
• It helps us understand how much energy is available to convert into motion.
• Understanding potential energy is essential to analyzing situations involving height, like sledding down a hill.