Paul and Kathleen start from rest at the same time at height h at the top of two differently configured water slides. The slides are nearly frictionless.
a. Which slider arrives first at the bottom?
b. Which slider is traveling faster at the bottom? What physical; principle did you use to answer this?
To determine which slider arrives first at the bottom of the water slides, we need to consider the properties of the slides and apply some basic physics principles.
a. Let's break down the problem into two scenarios:
1. Slide Configuration: The first slide is steeper and has a shorter length, while the second slide is less steep and longer.
To figure out which slider arrives first, we can analyze the effect of the slide configuration on their descent. In general, when the height difference is the same, a steeper slide will result in a higher acceleration, which means the object will move faster.
Assuming both slides have the same height difference h, since the first slide is steeper, the object on that slide will experience a greater acceleration compared to the object on the second slide. This means that the slider on the first slide will reach the bottom faster than the one on the second slide.
2. Slide Configuration: Both slides have identical steepness and length.
In this scenario, both sliders will experience the same acceleration, as the steepness and length of the slides are identical. Therefore, both sliders will arrive at the bottom simultaneously.
b. Now, let's consider the velocity of the sliders at the bottom:
To determine which slider is traveling faster at the bottom, we can apply the principle of conservation of mechanical energy. When neglecting any energy losses due to friction or air resistance, the total mechanical energy of a system remains constant.
At the top of the slide, both sliders start from rest, so their initial kinetic energy is zero. However, they both possess gravitational potential energy due to their initial height above the ground, represented by mgh (mass x gravity x height).
As the sliders descend, their potential energy is converted into kinetic energy. According to the conservation of energy, the sum of their kinetic and potential energies should remain constant throughout the motion.
At the bottom of the slide, both sliders reach the same height above the ground (zero), which means that their gravitational potential energy becomes zero as well. Therefore, the entire potential energy at the top of the slide is converted into kinetic energy at the bottom.
Since both sliders have the same initial potential energy (mgh) and the same final kinetic energy, their speeds will be equal at the bottom, assuming no external forces are acting upon them.
In conclusion, when the slides have different configurations, the slider on the steeper slide will arrive first at the bottom. However, when the slides have identical configurations, both sliders will reach the bottom simultaneously. Additionally, at the bottom, both sliders will have the same speed, as their gravitational potential energy is fully converted into kinetic energy.