John has a bow and arrow. He has found that the farthest he can shoot it is 400 feet. But he can only reach that distance if he aims the arrow halfway between straight up and straight at the target. In other words, he aims 45 degrees into the air.

What John would like to know is: how fast is the arrow going?

His friend Simon pulls out his smartphone and brings up a stopwatch program. Simon times how long it takes the arrow to travel 400 feet. The result is 4.988 seconds. Dividing 400 feet by 4.988 seconds, John gets an average velocity of 80.2 feet per second, or 55 miles per hour.

Now they know what the average velocity of the arrow was. But the arrow did not just go horizontally. Initially, at launch time, as it went forward, it also went up the same amount (as a result of the 45 degree angle of launch).

The true speed of the arrow at launch time must be more than 80.2 feet per second. In fact, since it is aimed at 45 degrees, it is moving up as fast as it is moving forward. They use the Pythagorean theorem to calculate the true launch speed of the arrow. The actual speed at launch time is a little over 113 feet per second, or 77 miles per hour.

They notice that the arrow goes up for a while, then seems to stop, and then it starts coming down. Gravity is making the arrow slow down, stop, and then speed up again. They wonder: how high does the arrow go?

Simon climbs up a tall ladder and drops the arrow while John works the stopwatch. He keeps climbing until the stopwatch says the arrow took one full second to hit the ground. They measure the height, and find it is 16 feet. The arrow fell 16 feet in one second.

The speed of the arrow started at zero. The average speed was 16 feet per second. What was the final speed? To get an average, we add the initial speed and the final speed and divide by two. Since the initial speed was zero, the final speed must be twice the average speed. The final speed is 32 feet per second.

In one second, the arrow sped up from zero to 32 feet per second. In the next second, it will speed up by another 32 feet per second. They have discovered that the acceleration of gravity is 32 feet per second per second.

The arrow starts off with a speed of 80.2 feet per second in the upward direction. Each second, gravity steals 16 feet per second  of that speed. We say that mathematically as \(16t^2\), where the variable \(t\) stands for time.

If we want to know the height of the arrow at any horizontal distance from the archer, we can first calculate the height if there were no gravity, and then subtract the effect of gravity from that.

Without gravity, the path of the arrow is a straight line. Using 45 degrees as the launch angle, it is a straight line with the slope of 1. That means that for every foot the arrow moves forward, it moves one foot upwards. Then we subtract \(16t^2\).

But wait! We don't know how many seconds the arrow has been flying. All we know is how many feet from the archer it is, horizontally (along the ground). But we know the horizontal velocity is 80.2 feet per second, so we can divide the distance by that to get the seconds. That looks like \( distance \over 80.2 \). When we replace \(t\) with that, we get \( 16 ({distance \over 80.2})^2 \).

We now have a formula for the height of the arrow as a function of the distance from the archer.

So now our equation is \(height = f(distance) = distance - 16({distance \over 80.2})^2\).

Up to this point, John and Simon have used only simple arithmetic to solve their problems. Nothing but addition, subtraction, multiplication, and division.

But to do that, they have limited themselves to an angle of 45 degrees. To make their equation more general, some elementary trigonometry can be added, and we can then use any angle we want.

Remember the first term was the height if there were no gravity. We knew that was the same as the horizontal distance because we were using 45 degrees. To get the height for any angle, we multiply the horizontal distance by the tangent of the horizontal distance. The tangent of an angle is the vertical change divided by the horizontal change, which is exactly what we need.

In the second term, we divided the distance by the horizontal velocity to get seconds. But we only know the horizontal velocity is 80.2 feet per second if the angle is 45 degrees. We know the actual velocity, however, and we can calculate the horizontal velocity for any angle by multiplying the actual velocity by the cosine of the angle. The actual velocity is the hypotenuse of a triangle, and the horizontal velocity is the bottom of the triangle, and the cosine is the ratio of the hypotenuse to the bottom.

Our new generalized equation now looks like this:

    \( height = f(distance) =  {{distance}  {tan(angle)}} - 16({{{distance}} \over {{velocity}   {cos(angle)}}})^2 \)

Now John and Simon can plug in any angle and any distance, and calculate the height, the number of seconds the arrow has been in the air, the horizontal speed, and the vertical speed. The graph below allows you to slide the numbers for the distance and the angle and see the results.

John notices that the angle of the arrow seems to follow the curve in the graph. He can think of the arrow as a line that touches the curve at a single point. Such a line is called a tangent. The slope of the tangent is the angle of the arrow. He can see that the slope of that line is the same as the ratio of the vertical speed to the horizontal speed. If he can find the slope of that line, he can save a lot of time calculating. He can get the vertical speed just by plugging in the horizontal speed, which was the very first thing he was able to calculate.

To find the tangent at any point on a curve, we can draw a line from that point to any other point on the curve. Then gradually move that second point along the curve towards the first point. The slope of the line gets closer and closer to the slope of the actual tangent to the curve at the first point.

Play with the sliders in the graph below. The distance of interest is the green dot, the tangent whose slope we wish to know. Move the red dot around, and look at how our estimate of the vertical speed gets closer to the actual vertical speed as the dots get closer together.

A problem arises when the dots are at the same place. The slope of the line is the vertical difference divided by the horizontal difference. But when the horizontal difference is zero, we can't do the division. We can estimate the slope this way, but when we actually get to the point of interest, the calculation fails us.

This is actually the problem calculus was invented to solve.

We are introducing here the concept of a limit.

We can ask ourselves if the slope of the line seems to approach a limiting value as the distance between the dots approaches zero.

We have a nifty mathematical notation for this concept:

$$\lim_{distance\rightarrow 300} f(distance) = -40.1$$

We read this as the limit as distance approaches 300 of \(f\) of distance is -40.1.

In our attempt to find the tangent of the curve at a point, we can make the formula much simpler by describing the problem in terms of functions.

The slope of the line is

$${y_1 - y_0} \over {x_1 - x_0}$$

But we can define \( y_0 \) as \( f( x_0 )\) and \( y_1 \) as \( f( x_1 ) \). This lets us express the slope of the line as

$${f(x_1) - f(x_0)} \over {x_1 - x_0}$$

Now let's make the formula even simpler by making a variable \( d \) equal to \( {x_1 - x_0} \):

$${f(x_0 + d) - f(x_0)} \over { d }$$

We can make that second point get closer and closer to the first point by making \( d \) smaller and smaller. Now we can express this concept as a limit:

$${\lim_{ d \rightarrow 0 }} {{ f( x_0 + d ) - f( x_0 ) } \over d } $$

We can see that as the difference between the x values (\( d \)) gets closer to zero, the value of

$${{ f( x_0 + d ) - f( x_0 ) } \over d } $$

gets closer and closer to the slope of the line at \( (x,f(x_0)) \).