Question Video: Finding the Sum of the Kinetic and Potential Energies of a Body Falling Vertically at a Given Point Mathematics

Video Transcript

A body of mass 20 kilograms fell from a height of 42.3 meters above the surface of the ground. Find the sum of its kinetic energy and its potential energy relative to the ground two seconds after it started falling. Take 𝑔 equal to 9.8 meters per square second.

We will begin this question by drawing a diagram. We know that the body begins a height of 42.3 meters above the ground. We need to calculate the information about the body after two seconds. We know that the initial speed 𝑢 is zero meters per second. We can use the equations of uniform acceleration known as the SUVAT equations to calculate the displacement and velocity after two seconds. As already mentioned, our initial velocity is zero meters per second. The time 𝑡 is equal to two seconds. And we know that acceleration due to gravity is 9.8 meters per second squared.

To calculate the velocity, we can use the equation 𝑣 is equal to 𝑢 plus 𝑎𝑡. Substituting in our values, we have 𝑣 is equal to zero plus 9.8 multiplied by two. This gives us an answer of 19.6. The velocity of the body after two seconds is 19.6 meters per second. The displacement 𝑠 can be calculated using the formula 𝑠 is equal to 𝑢𝑡 plus a half 𝑎𝑡 squared. Substituting in our values, 𝑠 is equal to zero multiplied by two plus a half multiplied by 9.8 multiplied by two squared. This is also equal to 19.6. The displacement of the body is therefore equal to 19.6 meters. After two seconds, that body has fallen 19.6 meters.

We can now include these values onto our diagram. We can work out the height of the body from the ground at this point by subtracting 19.6 from 42.3. This is equal to 22.7. After two seconds, the body is 22.7 meters from the ground. We are asked in the question to calculate the kinetic energy and the potential energy relative to the ground and then find their sum. We know that the kinetic energy of a body is equal to a half 𝑚𝑣 squared and the gravitational potential energy is equal to 𝑚𝑔ℎ. As the mass of the body is 20 kilograms, the kinetic energy after two seconds is equal to a half multiplied by 20 multiplied by 19.6 squared. This is equal to 3841.6.

Our units for energy is joules. Therefore, the kinetic energy is 3841.6 joules. To calculate the GPE, we need to multiply 20, 9.8, and 22.7. This is equal to 4449.2. The potential energy relative to the ground after two seconds is 4449.2 joules. The sum of these two values is 8290.8 joules.