A bird flying over the sea has a weight of 15 newtons and has a constant 765 joules of gravitational potential energy. How far above the sea does the bird fly?
In this question, we know we’ve got a bird flying over the sea. And it has a weight of 15 newtons. It also has a constant 765 joules of gravitational potential energy. What we’re asked to do is to find out how far above, that is, the height above the sea, that the bird is flying at.
So here’s the sea. Here’s what is meant to look like a bird. And this is the height at which it’s flying. We’ll call this ℎ. Now we’ve been told in the question that the bird has a weight 𝑊 of 15 newtons. We also know that it has a constant gravitational potential energy — we’ll call this 𝐸 sub 𝑔 — of 765 joules.
We can now look at the formulas for weight and gravitational potential energy to help us solve this problem. Firstly, let’s look at weight. The weight of an object 𝑊 is defined as the mass of that object 𝑚 multiplied by the gravitational field strength 𝑔.
Secondly, the gravitational potential energy. The gravitational potential energy of an object is defined as the mass of the object 𝑚 multiplied by the gravitational field strength 𝑔 multiplied by the height at which the object is, ℎ. This height is usually measured relative to the surface of the Earth, and in this case relative to the sea.
Now what we can notice is that the gravitational potential energy equation has an 𝑚𝑔 in it. This looks exactly like the 𝑚𝑔 in weight. Therefore, we can replace the 𝑚𝑔 in the gravitational potential energy equation with weight. So the gravitational potential energy of the object ends up being the weight 𝑊 multiplied by its height ℎ.
In this question, we’re actually trying to find out this height. So we need to rearrange this equation. We can divide both sides by 𝑊. And the 𝑊s on the right-hand side cancel, leaving us with 𝐸 sub 𝑔 over 𝑊 is equal to ℎ. So all that remains is to plug in our values. We get 765 joules divided by 15 newtons is the height at which the bird is flying.
By the way, earlier on in the question, we’re told that the gravitational potential energy is constant. All this means is that everything on the right-hand side of the equation is also constant. Of course the mass of the bird must be constant. It’s not like it’s gaining or losing weight as it flies. The value of 𝑔, the gravitational field strength of the Earth, is already a constant. And finally, the height of the bird must also be a constant.
So all this question is telling us is that the bird is flying at a constant height. And we’re trying to find out this height. Evaluating the fraction in our working out, we get a height of 51 meters.
Very quickly, how do we know that the answer that we’ve got is in meters? Well, we’re using standard units here. We know that the standard unit of weight is newtons. We know that the standard unit of energy is joules. And so our final answer will end up being in the standard unit for height, which happens to be a distance. And the standard unit for distance is meters. Therefore, our final answer is that the bird is flying at a height of 51 meters.