Assume that you take data on the projectile range for a variety of different launch angles A at a fixed initial velocity Vo. You would like to plot your data to see if the range formula discussed in the introduction is supported by the data. The easiest way to do this is to plot the range R on the y axis versus a quantity on the x axis that depends on the launch angle A and would result in a straight line. What quantity will you plot on the x axis to obtain a straight line under these conditions?

Go back to the earlier problem I just did.

"range = u (2 T)
= Vo cos T (Vo/4.9) sin T
= (Vo^2/4.9) cos T sin T "

plot range versus (cosT sinT)
which is
(1/2) sin 2 T

so plot versus sin 2T

But if you do that do T = 0 to 90 so 2T is from 0 to 180

Well, if you're looking to obtain a straight line when plotting the range R on the y-axis versus a quantity on the x-axis that depends on the launch angle A, you should use the sine of the launch angle A.

Why, you may ask? Well, it's because the range formula for projectile motion involves the square of the initial velocity V₀, so taking the square root of that will result in a linear relationship. And conveniently, the sine of an angle also ranges between -1 and 1, making it a perfect candidate for the x-axis quantity.

So, get ready to plot those ranges and sine angles, and watch that straight line form like a trajectorific masterpiece!

To obtain a straight line when plotting the range (R) on the y-axis versus a quantity on the x-axis, we need to find a quantity that depends on the launch angle (A) but results in a linear relationship with the range.

In this case, we can use the horizontal component of the initial velocity (Vo) as the quantity on the x-axis. This quantity, denoted as Vx, is the velocity in the x-direction and can be calculated using trigonometry.

The relationship between Vx and Vo can be determined using the launch angle A. Specifically, Vx can be obtained by multiplying the initial velocity Vo by the cosine of the launch angle A:

Vx = Vo * cos(A)

When the range R is plotted against Vx, the resulting graph will exhibit a linear relationship if the range formula discussed in the introduction is supported by the data. This is because the horizontal component of the velocity remains constant throughout the projectile's motion, resulting in a linear relationship between R and Vx.

Go back to the earlier problem I just did.