# Propagation of Error and Relative Error

## Homework Statement

2) A student is performing an angry bird experiment in real life. He calculates the range of the projectile by shooting the bird with a 30 degree angle and an initial velocity of v0 = 20 m/s.

a) If the relative accuracy of setting the projectile angle is delta theta/theta = 0.05, what would be the accuracy of the range estimation?

## Homework Equations

delta R= derivative of range equation*delta theta

R=v^2/g*sin(2*theta)

## The Attempt at a Solution

I determined the derivative of the range equation and got v^2/g*2*cos(2*theta). I then divided this by the original function and got delta x/x=2cot(2*theta)*delta theta. This is the point I am stuck at. Normally with the problems I have done so far I would get to this point and be able to substitute in the given value of delta theta/theta. However, the cot is throwing me for a loop and I just can't to find a way through it and past that point. All advice is appreciated. Thanks!

gleem
What is your problem with cotangent? You lost sight of what you are suppose to be looking for but you are on the right track.

What is your problem with cotangent? You lost sight of what you are suppose to be looking for but you are on the right track.

My problem is that if I put theta into the cot and solve the relative accuracy of the angle for delta theta and put that into the equation as well, I get a number over 100% which doesn't seem like it is even remotely correct and I just can't think of any other way to go about it.

gleem
What value are you using for dθ?

What value are you using for dθ?

The only value I could think of was to solve dTheta/Theta for dTheta. So the value is really 30*0.05.

gleem
That is you problem. Can you figure out why? Think units.

That is you problem. Can you figure out why? Think units.

I know that the answer I am getting currently would be in degrees which is incorrect but I can't seem to find a way to solve for dTheta without it encountering that issue. Is there another way to figure out dTheta?

gleem
dθ = .05⋅θ What units should you be using for θ? What other unit for angles is there?

dθ = .05⋅θ What units should you be using for θ? What other unit for angles is there?

Using radians I get a final answer of 0.0302. How do you know you need to use radians instead of degrees though. I know that the answer seems more reasonable now but in the future how can I tell?

jbriggs444
Homework Helper
Using radians I get a final answer of 0.0302. How do you know you need to use radians instead of degrees though. I know that the answer seems more reasonable now but in the future how can I tell?
If you take the derivative of a trig function the assumption is that the angle is expressed in radians. In those units, the first derivative of sine is cosine.

If you have a trig function and the angle is in degrees, you will need to insert a unit conversion factor of pi radians per 180 degrees each time you integrate or take a derivative.

It's usually easier to always do the math using radians and to convert the inputs and outputs as required.

gleem