An astronaut who has a mass of 80 kilograms is in a space station that orbits Earth in free fall. The astronaut is floating in a room of the space station, touching a wall that is perpendicular to Earth’s surface with her hand. She pushes against the wall with a force of 20 newtons. What is her acceleration perpendicular to the wall?
Alright, so we’ve got an astronaut who has a mass of 80 kilograms. She’s in a space station that orbits Earth in free fall. We also know that the astronaut is floating in a room of the space station touching a wall that’s perpendicular to Earth’s surface with her hand. She then pushes against the wall with a force of 20 newtons. We need to find out her acceleration perpendicular to the wall.
So let’s draw a quick diagram. Let’s say that this here is the Earth. Which means that now we’ve gotta draw a space station in orbit around the Earth. Well, if the space station is orbiting the Earth, then it must be undergoing circular motion which is why we’ve drawn the rather wonky dotted pink circle. We can record that when an object orbits the Earth in free fall, what that means is that the force holding the object in orbit is the gravitational force. And this force acts towards the center of the Earth which also happens to be the center of the circle. So gravity is providing the centripetal force necessary to keep the space station in orbit.
Now, let’s zoom in to the space station a little bit. So now we can see the space station a little bit more clearly, including the wall that’s perpendicular to the surface of the Earth. We know that the astronaut is touching that wall and is eventually going to push against it. And so, there’s our astronaut. We know that when she pushes against the wall, she pushes with a force of 20 newtons. We also know her mass, which is 80 kilograms. And we’re asked to calculate her acceleration perpendicular to the wall.
Now, let’s first discuss the force of gravity that we saw in the smaller diagram on the left. Well, we know that the force of gravity acts towards the center of the Earth, which is in this direction. Now, this force is directly perpendicular to the surface of the Earth. In other words, this force is parallel to this line that we’ve drawn here. Which means that the 20-newton force that the astronaut pushes with — let’s draw this again over here, 20 newtons, the gravitational force directly perpendicular as we’ve just said. We can also label this force as 𝑓 sub 𝑔, just the force of gravity, so we know exactly which one we’re talking about. So to reiterate, the 20-newton force is directly perpendicular to the force of gravity. That means that there’s no component of the gravitational force in this direction, the direction in which the astronaut applies the force. In other words, in that direction, we do not need to worry about any other force apart from the 20-newton force that the astronaut applies.
Luckily for us, we’re asked to calculate the acceleration of the astronaut perpendicular to the wall. That, once again, is this direction, perpendicular to the wall. Essentially, what this means is that we can completely ignore the force of gravity. We don’t need to think about it at all. The only force we need to worry about is the force that the astronaut applies against the wall. Now, we’re asked to work out the acceleration of the astronaut, not of the space station when she pushes against it. But, we’re only given the force that she applies on the space station. Happily, however, we can work out the force applied to the astronaut that causes her to accelerate. We do this using Newton’s third law of motion.
Newton’s third law states that if an object, let’s call it object one, exerts a force on another object, object two, then object two exerts an equal and opposite force on object one. In this case, our astronaut is object one. She’s exerting a force on the space station which is object two. Therefore, the space station exerts an equal and opposite force on the astronaut, 20 newtons to the left, as we’ve drawn it. Which means that using Newton’s third law, we’ve calculated the force on the astronaut. And therefore, we can find out her acceleration.
We know her mass as well as the force exerted on her body now. So we need to find a relationship that links together her mass, the force exerted on her, and her acceleration. We could do this using Newton’s second law. This states that 𝑓 is equal to 𝑚𝑎. Or, the force exerted on an object is equal to the mass of that object multiplied by its acceleration. The force, the mass, and the acceleration, it’s important to mention the standard units of each one of these quantities. The standard units of force is newtons, the standard units of mass is kilograms, and the standard unit of acceleration is meters per second squared.
Therefore, if we want to find the astronaut’s acceleration in meters per second squared, then we need to provide the equation with her mass in kilograms and the force in newtons. Luckily, this is exactly what we have. We have a 20-newton force and an 80-kilogram mass. So we can crack on with finding the solution. First, we need to rearrange the equation. We can do this by dividing both sides of the equation by the mass 𝑚. The masses on the right-hand side cancel, leaving us with: the force divided by the mass is equal to the acceleration. Now, all that remains is to plug in our values.
The acceleration is equal to a 20-newton force divided by the 80-kilogram mass, which leaves us with an acceleration of 0.25. And we’ve mentioned the units already. So our final answer is that the acceleration of the astronaut perpendicular to the wall is 0.25 meters per second squared.