A body of mass nine kilograms fell vertically from a point 3.4 meters above the ground. At a certain moment, the speed of the body was 3.9 meters per second. Determine the change in the body’s gravitational potential energy from this point until it reached a point 68 centimeters above the ground. Take 𝑔 equal to 9.8 meters per square second.
We will begin by drawing a diagram. We know that the body starts 3.4 meters above the ground. We know that the initial speed was zero meters per second and that at a certain moment the speed was 3.9 meters per second. We will begin by calculating the displacement 𝑠 at this point. We know that the acceleration due to gravity is equal to 9.8 meters per square second. This means that we can use the equations of uniform acceleration known as the SUVAT equations. 𝑢 is equal to zero; 𝑣 is equal to 3.9; 𝑎 is equal to 9.8. And we wish to calculate the value of 𝑠.
We will use the equation 𝑣 squared is equal to 𝑢 squared plus two 𝑎𝑠. Substituting in our values, we have 3.9 squared is equal to zero squared plus two multiplied by 9.8 multiplied by 𝑠. 3.9 squared is equal to 15.21, and the right-hand side simplifies to 19.6𝑠. Dividing both sides of this equation by 19.6 gives us 𝑠 is equal to 0.776 and so on. It is important to not round our answer at this point. When the velocity of the body is 3.9 meters per second, it has fallen 0.776 and so on meters.
We are interested in the point where the body is 68 centimeters above the ground. There are 100 centimeters in a meter, so this is equal to 0.68 meters. We want to calculate the gravitational potential energy between these two points. Gravitational potential energy is equal to 𝑚𝑔ℎ. We know that the mass of the body is nine kilograms and gravity is equal to 9.8 meters per square second. The value of ℎ will be the difference between our two positions.
As the initial height was 3.4 meters from the ground, 0.68 plus ℎ plus 0.776 and so on must be equal to 3.4. We can then calculate ℎ by subtracting 0.68 and 0.776 from 3.4. This gives us ℎ is equal to 1.9439 and so on. This is the height that the body travels between the two points. Multiplying this by the mass and gravity gives us 171.459. The gravitational potential energy between the two points is 171.459 joules.