I need help with a basic physics problem?

Question

Okay, in my physics class we are working on launching projectiles at an angle. I'm doing a lab that involves launching an object at different angles, but maintaing the same launch spped. I've discovered that the angle that gives the greatest range is 45 degrees, but I don't understand why it is that way.

Also, changing the launch speed but not the angle produces a graph that slants upward. So the mathmatical relationship should be linear, correct??

Answer 1

Range=Vo^2sin(2theta)/g

Contrary what some may have you believe, you do not need calculus for this. Just a little bit of reasoning skills. The range of the sin function is between -1 and 1. Thus the maximum value of sin(2theta) is 1.
sin(theta) is equal to 1 when theta is equal to 90 degrees. Since it's sin(2theta), then theta must be 45 degrees for the sine portion to equal 1. Thus the maximum range for a given initial velocity is achieved when the angle is 45 degrees.

As a side note, this equation is only valid when the projectile returns to its initial height. Do not use this equation on a test if the projectile lands higher or lower than its initial height.

Answer 2

Its the best angle because its exactly in between the two extremes, 0 degrees and 90 Degrees

I assume you produced a graph of velocity/ time, if the projectile doesn't increase or decrease in speed then yes it would be linear.

However, the I don't think you shot your projectile in an area without gravity... there for there should be a curve, in the shape of a parabola...which makes the mathematical relationship parabolic

Answer 3

If you are familiar with calculus, you could compute the angle which gives the maximum distance. You write an expression for distance vs angle, and take the first derivative, set it to zero, and solve for the angle. If you have not had calculus yet, then you can determine this experimentally only.

If you think you can follow it, here is the full derivation:
http://img140.imageshack.us/img140/3774/maxdistancox6.png

By the way, as you can see, the relation is that distance is proportional to the square of the velocity. Yes, it slants upward, but is not linear.