Calculus versus Algebraic Kinematic Equations

Question

Deriving the first equation of motion using either algebra or calculus you get the exact same thing, but my knowledge was that you use integral calculus in physics when you have a varying variable, while algebra is dealing with someone thing constant? Can someone please shed some light on what is algebra and calculus are telling me when I derive this equations?

Algebra:

$$ a= \frac {\Delta V}{\Delta t} $$

Rearrange, simplify:

$$ V_f = V_o +a \Delta t $$

Using Calculus:

$$ a = \frac {dV} {dt} $$

$$ \int_{t_i}^{t_f} {a dt} = \int_{v_i}^{v_f} { dv} $$

Integration:

$$ V_f = V_o +a \Delta t $$

Special relationship?

Answer 1

The "First Equation of Motion" you define is perhaps more accurately called the "First Equation of Motion with Constant Acceleration."

One would need to use Calculus to calculate the change in velocity when Acceleration is not constant, but what you call a "varying variable."

Your first equation which you arrive at by Algebra: $$V_f = V_o +a \Delta t$$

yields the correct change in velocity because the change in velocity $\Delta V$ over any interval of time $\Delta t$, large or small, is always $$\Delta V = a\Delta t$$ because a is always the same value.

When a is a "varying variable," a is different for different times, and therefore $\Delta V$ is different for different intervals of time $\Delta t$.

We can use calculus to find what $\Delta V$ is in this case by splitting $\Delta t$ into a very large amount of small times $dt$, and adding together the many very small resulting changes in velocity $dv = a(t)dt$. We add up the very small changes in velocity $a(t)dt$ between two points in time by writing: $$\Delta V =\int_{t_i}^{t_f} {a(t)dt}$$

Where the integral symbol is an elongated S symbolizing "sum."

I hope this helps you see when one must use Calculus in Physics.

Answer 2

If you want to derive that equation of motion in a serious way you always use calculus.

You start from the definition:

$a(t)= \frac {dv(t)}{dt}$

You integrate both sides from 0 to $t_f$

$\int_{0}^{t_f} {a(t)dt} = \int_{0}^{t_f} {v'(t)dt}$

and you finally get:

$v(t_f) = v(0)+\int_{0}^{t_f} {a(t)dt}$

If $a(t)=ct$ then

$v(t_f) = v(0)+ at_f$

Algebra is used in high schools where students do not know yet calculus but the derivation of a motion equation using algebra is considered more a trick than serious math.