Calculus – deriving the equations for simple projectile motion

We have worked several examples of simple projectile motion – meaning that the acceleration of gravity was constant and vertical, and there was no acceleration in the horizontal direction. (In particular, there is no air resistance.)

I simply handed us four equations and used them in a firstsecond… and third post. I said I would show how to derive them.

It was junior year in high school that I learned the equations for position and speed as a function of a constant acceleration. I didn’t take calculus until I was a college freshman… and at some point I decided that I knew enough calculus to derive the equations I had been told to memorize two years before.

This is about as elementary as it gets in calculus, but when it was all new to me, it was a thrill to see what it could do for me in physics. I will actually derive them in two slightly different ways.

Here we go.

First Derivation

Acceleration is the derivative of velocity…

\frac{d}{dt}v(t) = a(t)

and that means velocity is the integral of acceleration. We want the acceleration to be constant, so we write a instead of a(t):

v(t) = \int a \, dt

and do the integral, supplying a constant of integration when the integral sign disappears:

v(t) = a\ t + c .

It is convenient to evaluate the constant at time t = 0, because the term involving t drops out:

v(0) = a*0 + c

v(0) = c

and so

v(t) = a\ t + v(0) .

I learned that in reverse order, and – if you learned this equation – you probably did, too:

v(t) = v(0) + a\ t

“V equals v-zero plus a t.”

(I don’t see equations, I hear them.)

Next, velocity is the derivative of position…

\frac{d}{dt}s(t) = v(t)

so position is the integral of velocity. That is,

s(t) = \int v(t) \, dt

We plug in our formula for v(t)…

s(t) = \int (a\ t + v(0)) \, dt

We do the integral, getting another constant of integration…

s(t) = a\ t^2/2 + v(0)\ t + c .

Again, we evaluate that formula at t = 0 because two terms will drop out…

s(0) = a*0 + v(0)*0 + c

so

s(0) = c ,

and we have gotten

s(t) = a\ t^2/2 + v(0)\ t + s(0)

which I, at least, learned the other way:

s(t) = s(0) + v(0)\ t + a\ t^2/2

“S equals s-zero plus v-zero t plus a t-squared over 2.”

Second Derivation

There is yet another way to do this; use the fundamental theorem of integral calculus, which says:

if

\frac{d}{dt}F(t) = f(t)

then

F(b) - F(a) = \int_a^b f(t) \, dt .

(All that says is that the area under the curve f(t), between the points t = a and t = b, is given by the antiderivative F(t) evaluated at the points a and b. It relates the integral of f to the area under the curve f.)

In particular, we may take a = 0 and b = t.

F(t) - F(0) = \int_0^t f(\tau) \, d\tau .

(Yes, I have changed the variable of integration. I’m using “t” for the upper limit of a definite integral, and I cannot simultaneously use the symbol dt.)

Then from

\frac{d}{dt}v(t) = a(t)

with F = v and f = a, we get

v(t) - v(0) = \int_0^t a(\tau) \, d\tau .

For constant acceleration, a(\tau) = a\ , and we get…

v(t) - v(0) = \int_0^t a \, d\tau

which is…

v(t) - v(0) = a\ \tau\ |_0^t

v(t) - v(0) = a\ t - a*0 = a\ t

so finally

v(t) = a\ t + v(0) ,

just as before. This time we didn’t need to evaluate an unknown constant of integration; v(0) was there from the beginning, on the LHS.

Similarly, from

\frac{d}{dt}s(t) = v(t)

with F = s and f = v, we write the definite integral…

s(t) - s(0) = \int_0^t v(\tau) \, d\tau

Drop in our formula for v(t) – excuse me, v(\tau)\

s(t) - s(0) = \int_0^t (a\ \tau + v(0)) \, d\tau .

Do the integrals…

s(t) - s(0) = a\ \tau^2/2 |_0^t + v(0)\ \tau |_0^t

getting

s(t) - s(0) = a\ t^2/2 + v(0)\ t

and we may rearrange that, of course:

s(t) = s(0) + v(0)\ t + a\ t^2/2 .

Specializing to simple projectile motion

Finally, for simple projectile motion, we take the horizontal component of acceleration to be zero, and the vertical component of acceleration to be -g.

The horizontal equations, then, start with

s(t) = s(0) + v(0)\ t + a\ t^2/2

v(t) = v(0) + a\ t ;

With a = 0, the equations for the horizontal position and speed simplify to

s(t) = s(0) + v(0)\ t

v(t) = v(0) .

With a = -g, we get the equations for vertical position and speed:

s(t) = s(0) + v(0)\ t - g\ t^2/2

v(t) = v(0) - g\ t

And that’s all, folks.

4 Responses to “Calculus – deriving the equations for simple projectile motion”

  1. Jason Says:

    So im a bit late to you post, (a year and then some), but I came across this when preping for my final exam. I just wanted to drop a line and say you are very skilled in explaining these concepts in a systematic and clear way. It was a nice read and refresher. Bravo!

  2. Bryan Says:

    Agreed. Great post!


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