A 75-kilogram person jumps off a one-meter-high table onto the ground. If Earth has a radius of 6,370 kilometers, what is the acceleration of Earth due to the gravitational force between Earth and the person while they are in the air? Give your answer in scientific notation to two decimal places.
So here we have a table on Earth’s surface and a person jumping off the table. We know that the person will experience a force directed towards Earth’s center, which will cause them to fall back down to the ground, but also the earth experiences an attractive force towards the person because of their mass. And what we need to calculate is the acceleration of Earth due to the gravitational force between it and the person. Now we’re told that Earth has a radius of 6,370 kilometers and that the person has a mass of 75 kilograms. We’re also given the height of the table at one meter.
We’re calculating the Earth’s acceleration due to gravity. So we need to recall the equation for acceleration due to gravity, which is 𝑎 equals 𝐺𝑀 over 𝑟 squared, where 𝑎 is the acceleration due to gravity, which is the quantity we’re calculating. 𝐺 is the universal gravitational constant, which is 6.67 times 10 to the minus 11 meters cube per kilogram second squared. 𝑀 is the mass of the object we’re accelerating towards, which in this case is the person, which is 75 kilograms. And 𝑟 is the distance between the two objects’ centers of mass.
Now we’re told that the person is one meter above Earth’s surface, but Earth’s center of mass is located at Earth’s center, which is one earth radius away from the surface or 6,370 kilometers. Now we need this expressed in meters, so we need to recall that one kilometer is equal to 1,000 meters. And therefore, if 𝑟 is equal to 6,370 kilometers, this is equal to 6,370,000 meters. Now strictly, we’ve been told that the person is one meter above Earth’s surface, so we could add a meter to this. But note that we’ve only been given Earth’s radius to the nearest 10,000 meters. So if we were to add an extra one meter to this, it wouldn’t make any difference to the value at the accuracy we’re working. So in this case, we can just use the value of Earth’s radius, 6,370,000 meters.
We can now calculate the value of the acceleration of Earth, which is 𝐺𝑀 over 𝑟 squared or 6.67 times 10 to the minus 11 meters cube per kilogram second squared times 75 kilograms divided by 6,370,000 meters squared. This comes to 1.232 times 10 to the 22 meters per second squared, which is already in scientific notation. But we need to state the answer to two decimal places, which comes to 1.23 times 10 to the minus 22 meters per second squared. Now this is a very, very small number, which is what we would expect because we don’t routinely experience the Earth moving a great deal when we jump in the air.
So although the Earth does experience acceleration towards all the people and buses and everything else on its surface, it’s only a very, very small amount.