# What is the momentum between a light and heavy object in motion when the force applied is equal?

I was wondering, if a certain amount of force is used to push a very light object (like a ping pong ball) and an identical amount of force is used to push a very heavy object (like a steel ball), why does it seem to require much less force to stop the ping pong ball than is needed to stop the steel ball once they have begun moving?

Based on my studies, the difference seems to be related to momentum being very different between the 2 objects, but so far my understanding seems to suggest that their momentum should be similar. Since the light object should be moving very fast and the heavy object should be moving slowly, shouldn't their velocities balance out their momentum in the momentum equation of momentum = mass * velocity?

Thank you so much for reading my question and for your insight!

Your studies are correct, the momentums should be similar. However, the issue that is giving you trouble (and gives many people trouble) is that humans are very bad at estimating the forces they use to push things.

Consider the example of a ping pong ball (2.7g) and a steel ball of the same diameter (225g). Let's say you push the steel ball with enough force to reach 10m/s. (Chosen because that happens to be a reasonable speed for a ping pong ball serve). You'll find that if you push on the ping pong ball with the same force for the same time, the ping pong ball would have to be going 833m/s! That's over Mach 2!! I would comfortably say that stopping a ping pong ball coming at me at Mach 2 will be pretty difficult indeed!

So what's gives?

• If we throw a ball, we don't just accelerate the ball. We have to accelerate the hand and arm throwing the ball. In the case of the ping pong ball, this hand and arm are far more massive than the ping pong ball, so it takes a lot of force/energy to accelerate them. It takes the same amount of energy to accelerate the hand and arm when throwing a steel ball, but a larger portion of that force/energy goes into the ball.
• Air friction. Ping pong balls slow down quickly. This is always true, but especially true if the ping pong ball has broken the sound barrier ;-)
• Our intuition about force is typically horrid. There's a reason it took us until the 1600's to develop the math behind forces and accelerations. It turns out understanding force properly is not all that important for using the human body for what it is best at. The human body operates in very non-linear ways that have taken us 400 more years since Issac Newton and Gottfried Leibniz gave us the math behind modern physics. We're still figuring it out. No surprise, your intuition is very good at what the human body does, but not very good at what physics calls forces.
• Thank you so much for your amazing response! Your examples and descriptions were so incredibly helpful! I was wondering if I could pose a similar situation that has been puzzling me! Jan 2, 2017 at 6:25
• Sure, though if it's different enough from the question you just asked, it may be better to ask a second question and put a link to it here so that I can find it. (Stack Exchange benefits greatly from making questions and answer searchable) Jan 2, 2017 at 16:06
• Hi Cort, thanks again for the response! Here is the other question: physics.stackexchange.com/q/302390/140863 Jan 2, 2017 at 16:40

If you push 2 objects from rest with the same amount of force for the same amount of time, they acquire the same amount of momentum. This is Newton's 2nd Law : force x time = change in momentum.

Likewise it will take the same amount of time to stop the 2 objects if they are slowed down by the same force.