### Video Transcript

Two objects, ๐ด and ๐ต, are in deep space. Object ๐ด has a mass of 15,000 kilograms and object ๐ต has a mass of 26,000 kilograms. The distance between the centers of mass of the two objects is 25 meters. What is the acceleration of object ๐ต toward object ๐ด due to their gravitational interaction? Use a value of 6.67 times 10 to the minus 11 meters cubed over kilogram second squared for the universal gravitational constant. Give your answer in scientific notation to two decimal places.

So here are our two objects ๐ด and ๐ต. Weโre told that object ๐ด has a mass of 15,000 kilograms, so weโll call that ๐๐ด, and object ๐ต, weโre told, has a mass of 26,000 kilograms. So weโll call that ๐๐ต. Weโre also told that the distance between the centers of mass of the two objects is 25 meters. So letโs call that distance ๐. Weโre also told that the two objects are in deep space. This means there is nothing around that has any significant mass, and so the only forces we need to consider are those due to the mass of the two objects.

And we know from Newtonโs law of gravity that both objects will experience a gravitational force due to the mass of the other. Gravitational force always attracts, which means that this force will act to pull the two objects together. And the gravitational force always acts along the line connecting the objectsโ centers of mass. Because of this gravitational force, both objects will experience some acceleration towards the other. We could call the acceleration of object ๐ด ๐ sub ๐ด and the acceleration of object ๐ต ๐ sub ๐ต. And ๐ sub ๐ต is the one that weโre trying to find here.

Now we need to recall the relationship between a force and the acceleration due to that force. That is ๐น equals ๐๐, where ๐น is the force, ๐ is the mass of the object, and ๐ is the acceleration that object experiences due to the force. Now, in this case, weโre only asked to find the acceleration due to the gravitational interaction between the objects. So the force here is going to be the force due to gravity. So this force will be equal to the universal gravitational constant ๐บ times the mass of object one times the mass of object two divided by the distance between their centers of mass squared.

In this case, weโre trying to calculate the acceleration of object ๐ต. So the relevant mass here is that of object ๐ต, which weโve called ๐ subscript ๐ต. And that will be multiplied by the acceleration of object ๐ต, which is ๐ subscript ๐ต. And this is equal to the universal gravitational constant times the masses of the two objects, which weโve called ๐ subscript ๐ด and ๐ subscript ๐ต, divided by the distance between their centers of mass, ๐ squared.

The first important thing to note here is that the mass of object ๐ต appears on both sides, which means those cancel out. And weโre left with the acceleration of object ๐ต, which is equal to the universal gravitational constant ๐บ, times the mass of object ๐ด divided by the distance between them squared.

So putting numbers into this, we have ๐บ is equal to 6.67 times 10 to the minus 11 multiplied by the mass of object ๐ด, which is 15,000 kilograms, divided by the distance of 25 meters squared. And once weโve put all these numbers in, we get an answer of 0.0000000016008. And thatโs quite difficult to read, which is why weโre asked to give the answer in scientific notation, which means as a number between one and 10 times 10 to some power.

We can do that by moving the decimal point one, two, three, four, five, six, seven, eight, nine spaces, which gives us 1.6008 times 10 to the minus nine. And then weโre asked for two decimal places. So that becomes 1.60 times 10 to the minus nine.

Now for the units weโve used SI units everywhere. We had meters cubed over kilogram second squared for ๐บ, the mass of object ๐ด was given in kilograms, and the distance in meters, which means that our answer will be in the SI units of acceleration, which are meters per second squared. So the acceleration of object ๐ต due to its gravitational interaction with object ๐ด is 1.60 times 10 to the minus nine meters per second squared.

Now one important thing to notice about this problem is that we didnโt use the mass of object ๐ต at all because it canceled out in this equation. So that leads us to the important conclusion that the acceleration of an object due to gravity does not depend on its mass. It only depends on the mass of the object whose gravitational field itโs experiencing, in which case that was ๐๐ด, and its distance from that object, which means this acceleration would have been the same for any object of any mass placed 25 meters away from the center of object ๐ด.