# Why does force cause a change in acceleration rather than a change in mass?

$$F=ma$$, so force and acceleration are directly proportional. Force and mass are also directly proportional. Observationally, force alters acceleration and mass is constant but the equation itself does not dictate that the mass must be constant. Is the answer just that the math is meant to describe the observation and that's just how it is?

As a side note, when a new equation is discovered, how would you know what is variable and what is constant without just making an assumption?

• Just because an equation takes a certain form (force is the multiplication of mass and acceleration) does not mean all the mathematical things you can do with that equation are physical. E.g. in physics often when a sqrt is taken, we discard the negative result because the quantity in question (mass, speed) must be positive to describe the real world. Mass is conserved separately from momentum. Commented Jul 30, 2022 at 3:14
• "Is the answer just that the math is meant to describe the observation and that's just how it is?" Yes. Commented Jul 30, 2022 at 3:56
• Suppose you apply a zero force to your friend? What do you think will happen :your friend will accelerate with any rate because his mass will become zero?
– user326901
Commented Jul 30, 2022 at 9:34
• Trivially as it may seem, different masses may be put into the equation, which allows us to compare mass a to mass b. If that is what you do not mean with "change" - why do you not say so? I like this question because my answer to the latter is: because that would spoil what this question is able to teach me - that the answer yes to "math is meant to describe" is correct. If that's not been only insinuated, though... Thank you very much. Commented Nov 28, 2022 at 15:21

It actually is implied in the formula you stated that mass is constant with time. Newton's original formulation for his 2nd law is:

$$\vec{F} = \frac{d\vec{p}}{dt}$$

Where $$\vec{p} = mv$$, i.e., momentum.

Putting it in the form of $$F = ma$$ implies that mass is constant with time.

About your second question. I imagine relationships between variables are supported by experimental evidence. You try and change only a single independent variable and see how the dependent variable changes as a result.

Alternatively, you can work with constants and variables already known to reformulate equations into more illuminating forms or tease out relationships. However, my impression is that purely theoretical results still require experimental evidence to be treated seriously by people.

• Thanks for the perspective! I simply learned F=ma but was unaware of Newtons formulation. On the second question, you reminded me that while working on new theory, you'll be drawing on previously known relationships with existing constants and variables that are already known, presumably from earlier experimentation. Commented Jul 30, 2022 at 5:54

Welcome to physics SE! $$F=ma$$ is not the fundamental form of Newton's second law, but force equals rate of change of momentum. To get $$F=ma$$we assume mass is constant. If you want to treat a problem with a varying mass like a rocket you start from the more fundamental form.

• Thanks for pointing out that force is rate of change of momentum. That helps to understand what is really being described. Commented Jul 30, 2022 at 5:58

$$F=ma$$, so force and acceleration are directly proportional. Force and mass are also directly proportional. Observationally, force alters acceleration and mass is constant but the equation itself does not dictate that the mass must be constant.

This is partially true - we observe that there is a quantity that is preserved and that this is a resistance to change in motion (mass). There are actually two notions of mass, inertial and gravitational. It's been observed that these two quantities are always equal, and these give us information about an objects willingness or unwillingness to change. This can help you understand why it might act as a constant - it's really weighting how much the motion can or can't change.

Is the answer just that the math is meant to describe the observation and that's just how it is?

I personally believe the math is the root of things in physics, not the other way around, so it's good to question what the math means. You can ask different types of physics enthusiasts to get different opinions.

As a side note, when a new equation is discovered, how would you know what is variable and what is constant without just making an assumption?

When you are looking into theory, you have to understand what each quantity "is", meaning what its form is. In $$F=ma$$, the theory is formulated such that $$m$$ is a constant, and $$a$$ is the acceleration vector, meaning $$F$$ must be a vector. You can also track things through the math. For instance, I just said that given that $$m$$ is a scalar, and $$a$$ is a vector, $$F$$ also must be a vector.

It's worth noting that actually what you are learning is the nonrelativistic limit of physics. If you look at more advanced and more complete forms of physics, mass actually can change, and this is treated in relativity courses where mass becomes more synonymous with energy. Also, in general relativity, acceleration is the root of forces, not the other way around.

• First, Thanks for taking the time to write a great answer. So you say, "a is the acceleration vector". My (limited) understanding is that a vector requires a magnitude and direction, so to me "a" appears to be a scalar quantity. Can you clarify? Commented Jul 30, 2022 at 5:32
• no problem, glad it was helpful. so yes, vectors do have magnitude and direction. for acceleration, the magnitude is the amount that velocity is changing, and the direction is the direction of that change. for example if you are in a car that is decelerating, say at a rate of 10 km per second, your car is getting slower by 10km every second, so the magnitude of this acceleration is 10km, and the direction is backward (assuming you are driving forward on a road). Note your velocity is still forwards, but the acceleration describes how the velocity changes. @Fëakhelek Commented Jul 30, 2022 at 5:40
• Also, I'm not taking physics classes now, I'm more of a dabbler. I don't really have the math background to go much further but I try to understand the concepts. For example, I recently learned to understand the 4 vector because in reading about relativity it seemed to me that we must be moving forward in the time dimension at the speed of light. I can't do the math but I grasp the idea, which is, to me, better than being able to do the math but not truly "grok" the concept behind it. Commented Jul 30, 2022 at 5:47
• that's awesome for sure but just remember sometimes the devil's in the details ;) best of luck with your endeavors @Fëakhelek Commented Jul 30, 2022 at 6:10