Question

Can someone give me a general example of an algebraic and a calculaic way to solve the same physics problem?

Answer 1

Yes!

Answer 2

If we start with the calculus, we can see how the algebraic equations of motion are derived.

Let's take a simple example of a man pushing a box along a perfectly smooth, flat surface. The box has mass of 10kg and he pushes with a force of 10N. This yields an acceleration of \(1 m/s^2\). Let's determine the velocity and distance traveled after 10 seconds.

We know that\[v = {dx \over dt}\] and \[a = {dv \over dt}\]Therefore, to find the velocity after 10 seconds, we use the following formula\[dv = a dt \rightarrow \int\limits dv = \int\limits_0^{10} a dt \rightarrow v = a (10 - 0) + v_0\]This equates to \(10 m/s\).

This would be solved from the algebra as \(v = at + v_0\). Remember our EOM with constant acceleration?

To find the distance traveled, we solve the following\[dx = v dt \rightarrow \int\limits dx = \int\limits_o^{10} v dt \rightarrow x = \int\limits_0^{10} (at + v_0) dt \rightarrow x = {at^2 \over 2} + v_0t + x_0\] You'll find the solution to this integral on the list of constal acceleration EOM too.

This example is trivial. It doesn't reveal any benefit of solving with the Calculus, since these equations are very well known.

How about we solve a problem where acceleration is NOT constant. Imagine the guy pushes the box with a force of \(F = 10t\) where t is time in seconds. From newton's first law, we see that \[F = ma \rightarrow 10t = ma \rightarrow a = {10 t \over m} \rightarrow a = {10 t \over 10} = t\]

From the calculus, \[a = {dv \over dt} \rightarrow t = {dv \over dt} \rightarrow dv = t dt \rightarrow \int\limits dv = \int\limits_0^{10} t dt \rightarrow v = {t^2 \over 2} |_0^{10}\]Again, for distance\[v = {dx \over dt} \rightarrow \left ( {t^2 \over 2} + v_0 \right) = {dx \over dt} \rightarrow dx = \left ( {t^2 \over 2} + v_0 \right) dt \rightarrow \int\limits dx = \int\limits_0^{10} \left ( {t^2 \over 2} + v_0 \right) dt\] \[x = {t^3 \over 6} + v_0t + x_0 |_0^{10}\]

The calculus is also very helpful in the case where mass changes, too.