How do kinematic equations work regardless of mass of the object?

Question

I came across this question (very simple):

"A dog is running and starts to get faster at $2 ms^{-2}$ for $3s$. If the dog covers $20 m$ over this time, what velocity did it start with?"

Using the kinematic equations, the answer is $3.7 m/s$. My teacher said that this is true regardless of the nature of the object. For example, if it were a ball that was accelerating at $2 ms^{-2}$ for $3 s$ and covered $20 m$, the answer would still be $3.7 m/s$.

Why? How come the mass does not affect it? I understand it intuitively, but I can't seem to find an answer after thinking hard about it. Shouldn't forces such as gravity alter the acceleration or motion of an object? And gravity depends on mass.

I understand that kinematics is the branch of physics not concerned with forces, but how is it so accurate (provided constant acceleration)? How is it that when figuring out the motion through kinetics (which does depend on the object, e.g. its mass), you still get the same answers as if it were done through kinematics?

P.s. I am a high school student so i would appreciate it if the answers could be simple enough for me to understand (if it turns out to be complicated)

P.p.s I know calculus, basic derivitives, integrals, and up to first-order differential equations if it comes to that.

Much thanks!

Answer 1

how is it so accurate (provided constant acceleration)?

This is 100% it. We are assuming that the ball and the dog have the same acceleration (implicitly). This means we are a priori setting $F$ in $F=ma$ to whatever it has to be such that $a$ is the same for the difference mass objects. Concretely, if the ball is $1$ kg and the dog is $10$kg, then the force that is used to accelerate both of them to $a$ is different and the one for the dog is $10$ times larger. In kinematics, we don’t even care about this force, we are just given the value of the constant acceleration.

You mention differential equations, the kinematic equations are all useful forms of what happens when you consider the differential equation $$\frac{d^2x}{dt^2} =a$$ for constant $a$, which is simply the statement that acceleration is constant. The reason there are 5 SUVAT equations is because sometimes we want to solve for things in terms of different variables given in the problem, but they all come from this ODE.

How is it that when figuring out the motion through kinetics (which does depend on the object, e.g. its mass), you still get the same answers as if it were done through kinematics?

I am assuming you mean when you are given the force (or collection of forces on an object) and the mass and are expected to use Newton’s Second Law to find the acceleration. The reason this works is because it results in the same equation of motion, we obtain the same differential equation; the only extra step was that you were expected to find the acceleration rather than just have it as a given.

Answer 2

The kinematic equations are derived directly from the definitions of velocity and acceleration, assuming constant acceleration. Both of these variables do not contain any mass term, so the kinematic equations do not contain any mass term.