A body of mass 20 kilograms was projected vertically upward from the ground with a velocity of 30 meters per second. It passed by a point A when its velocity was 10 meters per second. Calculate the potential energy of the body at point A. Neglect the air resistance and consider the acceleration due to gravity to be 9.8 meters per second squared. Find the distance between point A and the ground surface in meters. Give your answer to one decimal place.
In this question, we are told that a body of mass 20 kilograms was projected vertically upwards with a velocity of 30 meters per second. When it passed a point A, its velocity was 10 meters per second. The first part of this question asks us to calculate the potential energy of the body at point A.
We will do this using the conservation of energy and by considering the change in kinetic energy and change in potential energy. We will let the initial kinetic energy be 𝐾 sub i and the final kinetic energy be 𝐾 sub f. The change in kinetic energy is therefore equal to 𝐾 sub f minus 𝐾 sub i. Likewise, we have a change in potential energy of 𝑃 sub 𝑓 minus 𝑃 sub 𝑖.
We are told to neglect air resistance, which means that the sum of these changes in energies must equal zero. Recalling that gravitational potential energy is equal to 𝑚𝑔ℎ, the initial potential energy will be equal to zero. As already mentioned, the sum of the changes in energies must equal zero. This means that the final potential energy must equal the initial kinetic energy minus the final kinetic energy.
And since kinetic energy is equal to a half 𝑚𝑣 squared, the final potential energy is equal to a half multiplied by 20 multiplied by 30 squared minus a half multiplied by 20 multiplied by 10 squared. This is equal to 9000 minus 1000, which equals 8000. The potential energy of the body at A is 8000 joules.
The second part of our question asks us to find the distance between point A and the ground. We will call this distance ℎ. And since the potential energy at point A is 8000 joules, 𝑚𝑔ℎ must equal 8000. Substituting in our values for 𝑚 and 𝑔, we have 20 multiplied by 9.8 multiplied by ℎ is equal to 8000. 20 multiplied by 9.8 is 196. And dividing through by this, we have ℎ is equal to 8000 divided by 196. Typing this into the calculator gives us 40.816 and so on. And rounding to one decimal place as required, the distance between point A and the ground surface is 40.8 meters.