We begin in this Chapter to study the motion of projectiles when they are governed by Galileo's gravitational law. A simple first question to ask is this. Assume that a projectile is launched vertically (straight up) at an initial height

with an initial vertical velocity

. How does one express the height:

as a function of time.

Now we saw when we looked at position and velocity portraits, that the derivative of

is interpreted as the instantaneous velocity of the projectile at each instant. And according to Galileo's law, if we measure time in seconds and distance in meters, gravity subtracts 9.8 meters/sec from the velocity each second. Thus since at time 0, we assumed the upward velocity was

meters/sec, the velocity at any later time (until the projectile is stopped by something) will be

measured in meters/sec. Let us represent the height by

as a function of time and represent velocity by v(t).

This is because at

, the height will be

, and the derivative with respect to

is what we want:

. If this is not clear, you should review the formula for calculating the derivative of a polynomial in any standard text.

But this is only a guess because as far as we know, there are other functions that satisfy the same conditions. This procedure: finding a function whose derivative is a given function is called "anti-differentiation" or "indefinite integration." We will see more on this in in Satellite Orbits where we show with Rolle's Theorem the justification for our guess. This

is in fact the only differentiable function that satisfies both conditions. You will see how Rolle's theorem allows us to conclude that.

You may check this formula in the microworld by choosing an initial height, initial velocity, and launching a projectile straight up (at a 90 degree angle with the ground). Then record the maximum height attained, and the peak time (the time required to reach that height) and use the calculator to see if you got it right. We show you how to do that now.

On this page, you may experiment with ballistic projectiles to discover how high they rise in the air depending on three things:

Cannon height: This is the altitude above the ground at which the cannon is placed. The height is measured in meters, and you determine this height by adjusting the slider at the upper right:

Initially, the height is 0, or ground level. As you move the slider, the height changes. To make fine adjustments to the slider value, click on the thumb, and then press the left or right arrow keys on the keyboard. Finally, you may set an absolute value by hand if you right click on the slider to get its menu, then select Settings, and type in the Current Value desired.

Cannon speed: This is the absolute speed in

with which the ball leaves the cannon. Initially, it is set to

As you change the speed the size of the cannon (at the lower left of the Graph2D) changes to reflect that.

Cannon angle: This is the angle, measured in degrees, that the cannon makes with the ground. Initially it is set to 30 degrees, and as you move the slider between 0 and 90 degrees, you will see the cannon rotate to reflect the change. If you choose an angle larger than 90 degrees, the system will use 90 degrees.

You may fire the cannon at will by pressing the Fire Cannon button. If you check the Sound? checkbox, then you will hear it fire. To clear the screen, press Clear.

which is scaled from -50 to 500 meters in each direction. You may also zoom-in around a rectangle that you draw by pressing the In button, then clicking the mouse on the upper-left corner of the new window and dragging it to the lower right corner, and releasing it. You may do this repeatedly, and to zoom out one step, just click Out. To zoom back to the start, click Out repeatedly. If you check Trace? then the coordinates of the mouse will be reported in the adjacent field as you move it around.

You are told the maximum height attained in meters, and the time required to attain that height (peak time) in seconds.

Now you may test your formula in the following way. Go to the calculator. You will find general instructions for its use there. Just press

and define a function height of three variables t (for time), h (for altitude) and v (for speed) that you think might be correct. You may use the constant g for the gravitational acceleration (9.8 meters per second per second) in your formula.

For example, you we might guess (incorrectly) that

Define the function of three three variables by typing the definition in the yellow command field, then pressing Enter.

make height(t,h,v) h + v*t -g*t^2; (Enter) or

height(t,h,v) := h + v*t -g*t^2; (Enter)

Now a simulation with the angle set to 90 degrees and the speed set to 50 with altitude 0 gives the result:

So we test the formula by calculating:

Calc height(5.1020408, 0, 50); (Enter)

The system returns the answer 0.0000008 in the white MathEdit Window. Something is definitely wrong with this formula. We should have found something close to 127.5510204.

Clear the Graph2D whenever you like by typing

clear; (Enter)

When a projectile is launched, it generally has two components of velocity at the moment of launch: a vertical component, and a horizontal component. If, for example, the projectile made an angle with the horizontal, and had initial speed then the horizontal component of velocity initially would be: and the vertical component of velocity would be .

Suppose the initial speed

was given as constant, and the initial height:

. And suppose you could "adjust" the angle

. Here,

Let us first write a formula that expresses the greatest height,

attained, as a function of

. For the greatest height, we observe that it is attained when the vertical velocity is 0. This happens when

Now, write a formula that expresses as a function of

the amount of time,

, the projectile is in the air. This is a bit more difficult.

This means

Let us write for the gravitational acceleration. So Then our equation becomes

Solving, we get

We can discard the negative sign, since it corresponds to a time before launch that is meaningless for our problem. It is in fact the time at which the projectile must be launched from the ground to arrive at the height 10 with the upward velocity V�sin(q) ). So our answer is

You can check this answer by typing (for example, for angle 30 degrees, and

). We will supply the angle in degrees, and so we apply the built-in radian function to convert that to the radians that the trigonometric functions expect.

The system reports 5.4748058 seconds in the air. Our experiment with the same data reports that 2.5510204 seconds elapse until the peak time (maximum height is attained). Is this reasonable? Can you check it? On the next exploration page, the system will report the crash time, the actual amount of time the ball was in the air.

You can of course experiment with the maximum height attained on this page. Suppose for example, the cannon was placed at height 10 meters with a muzzle speed of 50 meters/sec and was turned to an angle of 30 degrees. Following our directions at the top of the page, set these things up.