Question Video: Determining the Velocity of a Body in Motion on an Inclined Plane with a Resistive Force Mathematics

Video Transcript

A car descended 195 meters on a slope from rest, which is equivalent to a vertical distance of 14 meters. Given that two-sevenths of the potential energy was lost due to resistance and that the resistance remained constant during the car’s motion, determine the car’s velocity after it had traveled the mentioned distance of 195 meters. Take 𝑔 equal to 9.8 meters per second squared.

Let’s begin by sketching a diagram to model the scenario. We are told that a car descended 195 meters on a slope from rest. This means that its initial velocity 𝑉 sub i is equal to zero meters per second. We are told this is equivalent to a vertical distance or height of 14 meters. Our aim in this question is to calculate the car’s velocity after it has traveled this distance. We will call this velocity 𝑉 sub f. We will answer this question using our knowledge of the conservation of energy. And we are told that two-sevenths of the potential energy was lost due to resistance. We recall that the potential energy of an object is equal to 𝑚𝑔ℎ, where 𝑚 is the mass of the object, 𝑔 is the acceleration due to gravity, and ℎ is the vertical height.

The kinetic energy of an object is equal to a half 𝑚𝑉 squared, where once again 𝑚 is the mass of the object and 𝑉 is its velocity. Recalling that two-sevenths of the potential energy was lost due to resistance, the change in kinetic energy will be equal to five-sevenths of the initial potential energy. We know that the change in kinetic energy is equal to a half 𝑚𝑉 sub f squared minus a half 𝑚𝑉 sub i squared. And this must be equal to five-sevenths 𝑚𝑔ℎ. We know that 𝑉 sub i is equal to zero meters per second, and the vertical height ℎ is equal to 14 meters. Dividing through by 𝑚, our equation becomes a half 𝑉 sub f squared is equal to five-sevenths multiplied by 9.8 multiplied by 14.

Next, we can multiply through by two such that 𝑉 sub f squared is equal to ten-sevenths multiplied by 9.8 multiplied by 14. We can then square root both sides of our equation. And since 𝑉 sub f must be positive, this is equal to the square root of 196, which is equal to 14 meters per second. The car’s velocity after it has traveled a distance of 195 meters is 14 meters per second.