Question Video: Finding the Kinetic Energy of a Body Falling Freely from a Given Height before It Reaches the Ground Mathematics

A body of mass 15 kg fell from a height of 15 m above the ground. Using the work energy principle, find its kinetic energy just before it hit the ground. Consider the acceleration due to gravity to be 9.8 m/s².

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Video Transcript

A body of mass 15 kilograms fell from a height of 15 meters above the ground. Using the work energy principle, find its kinetic energy just before it hit the ground. Consider the acceleration due to gravity to be 9.8 meters per second squared.

The work energy principle tells us that the net work done on an object is equal to the change in its kinetic energy. Work itself is the energy gained or spent when an object moves in the same direction or in the opposite direction to some external force. We can express this as a formula as work is equal to the force applied times the distance through which the object moves while the force is being applied. So our plan is to find the work done on the body that we’re interested in, equate this to the kinetic energy, and that will be the kinetic energy that we’re looking for.

Now that we know what we need to do, let’s draw a diagram to organize our information. Here we have the body with mass 15 kilograms and the ground that is 15 meters away. We’ve also labeled the force of gravity, which is the force pulling the body down toward the ground. In fact, the force of gravity is the only force acting on the body in this situation. So it is the only force that contributes to the total work. Okay, so the force in the work formula is going to be the force of gravity. But what about the distance? Well, this is the distance that the body traveled while the force of gravity was pulling on. The gravity is always pulling on the body, so it is acting on the body for the entire 15 meters from where the ball starts until it hits the ground.

So the work done by gravity is the force of gravity acting on the body times the 15 meters the body traveled. From Newton’s second law that relates force to mass times acceleration, we know that the force of gravity acting on the body is the mass of the body times the acceleration due to gravity, which we’ve represented with the letter 𝑔. We’re given that the mass of the body is 15 kilograms, and we’re told that the acceleration due to gravity is 9.8 meters per second squared. Substituting this expression for force of gravity in the work formula, we get 15 kilograms times 9.8 meters per second squared times 15 meters. We wrote this without units just to make it easier to see the multiplication. 15 times 9.8 times 15 is 2205.

In keeping with the work energy principle, we know that the final units should be units of energy. Since in our other quantities we used kilograms for mass, meters for length, and seconds for time, the appropriate unit of energy for this final quantity is joules. So the work done by gravity as the ball fell was in total 2205 joules. But by the work energy principle, this work done during the course of the fall is exactly the final kinetic energy just before the ball finishes falling. So the kinetic energy of the body just before it hits the ground is 2205 joules.

It’s worth noting that our final calculation of work was a product of the mass of the body, the acceleration due to gravity, and the height of the body. But this product is exactly the gravitational potential energy of the body when it is at a height of 15 meters. This is not a coincidence. As the body falls, its gravitational potential energy changes and work is just another way to think about changes in energy.

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