### Another differential equation: projectile motion

Here we encounter the fundamental idea that *if $s=s(t)$ is
position, then $\dot{s}$ is velocity, and $\ddot{s}$ is
acceleration.* This idea occurs in all basic physical science and
engineering.

In particular, for a projectile near the earth's surface
travelling straight up and down, ignoring air resistance, acted upon
by no other forces but *gravity*, we have
$$\hbox{ acceleration due to gravity }= -32 \hbox{ feet/sec }^2.$$
Thus, letting $s(t)$ be position at time $t$, we have
$$\ddot{s}(t)=-32.$$ We take this (approximate) *physical fact* as
our starting point.

From $\ddot{s}=-32$ we *integrate* (or *anti-differentiate*) once to undo one of the
derivatives, getting back to *velocity*:
$$v(t)=\dot{s}=\dot{s}(t)=-32t+v_o$$
where we are calling the *constant of integration*
‘$v_o$’. (No matter which constant $v_o$ we might take, the derivative
of $-32t+v_o$ with respect to $t$ is $-32$.)

Specifically, when $t=0$, we have
$$v(0)=v_o$$
Thus, the constant of integration $v_o$ is **initial velocity**. And
we have this formula for the velocity at *any* time in terms of
*initial* velocity.

We integrate once more to undo the last derivative, getting
back to the *position* function itself:
$$s=s(t)=-16t^2+v_ot+s_o$$
where we are calling the constant of integration
‘$s_o$’. Specifically, when $t=0$, we have
$$s(0)=s_o$$
so $s_o$ is **initial position**. Thus, we have a formula for
position at *any* time in terms of *initial position* and *initial velocity*.

Of course, in many problems the data we are given is *not* just the initial position and initial velocity, but something
else, so we have to determine these constants indirectly.

#### Exercises

- You drop a rock down a deep well, and it takes $10$ seconds to hit the bottom. How deep is it?
- You drop a rock down a well, and the rock is going $32$ feet per second when it hits bottom. How deep is the well?
- If I throw a ball straight up and it takes $12$ seconds for it to go up and come down, how high did it go?

#### Thread navigation

##### Calculus Refresher

#### Similar pages

- Exponential growth and decay: a differential equation
- An introduction to ordinary differential equations
- Ordinary differential equation examples
- Solving linear ordinary differential equations using an integrating factor
- Examples of solving linear ordinary differential equations using an integrating factor
- The idea of the derivative of a function
- Derivatives of polynomials
- Derivatives of more general power functions
- A refresher on the quotient rule
- A refresher on the product rule
- More similar pages