A helicopter of mass 2,630 kilograms descended vertically from a height of 250 meters
to a height of 150 meters. Find its loss in gravitational potential energy. Consider the acceleration due to gravity to be 𝑔 equals 9.8 meters per square second
and give your answer in scientific notation.
Remember, when we think about the gravitational potential energy of an object, we’re
thinking about the potential that object has to do work as a result of being located
at a particular position in a gravitational field. So let’s think about what’s happening to the helicopter.
Initially, the helicopter, which has a mass of 2,630 kilograms, is at a height of 250
meters. It then descends vertically to a height of 150 meters. We can therefore find its loss in gravitational potential energy by calculating its
gravitational potential energy at the start, 𝐸 sub 𝑔 sub one, and then calculating
its potential energy after its descent. Let’s call that 𝐸 sub 𝑔 sub two. And of course the formula we used to calculate gravitational potential energy is mass
times gravity times height. Now, when we work with kilograms, meters, and meters per square second, the units are
kilograms square meters per square second, but that’s equivalent to joules.
Now, in this question, we’re already given a mass in kilograms and we’re working with
heights in meters. So we can calculate the potential energy at the start by multiplying mass, 2,630, by
gravity, 9.8, by height, 250. That gives us 6,443,500 joules. Then, we can calculate the potential energy of the helicopter after its descent by
multiplying 2,630 by 9.8 by the new height, 150. And we see that the new potential energy is 3,866,100 joules. Then, the loss in gravitational potential energy is the difference between these, and
that’s 2,577,400 joules.
Now, we’re going to give our answer in scientific notation, in other words in
standard form. To do so, let’s round to two significant figures. That gives us 2,600,000. Then, we recall that to write a number in standard form, we give it in the form 𝑎
times 10 to the power of 𝑏. 𝑎 is greater than or equal to one and less than 10. And of course 𝑏 is an integer. This can be positive or negative, depending on the size of the number. For very large numbers, as in this case, 𝑏 will be a positive number.
So to ensure we have a number between one and 10, let’s take the first two digits,
two and six, and make 2.6. We would need to multiply 2.6 by 10 to the power of six to make it 2,600,000. So 2,600,000 must be equivalent to 2.6 times 10 to the sixth power. So the loss in gravitational potential energy is 2.6 times 10 to the sixth power
Now we might observe that this is in fact the same as multiplying the mass by the
gravity by the change in height. So we could have equivalently calculated 2,630 times 9.8 times 250 minus 150 or times
100. Either method is equally valid.