Propagation of Error and Relative Error

[GoCubs12]

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.

 

[GoCubs12]

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?

 

[GoCubs12]

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.

 

[GoCubs12]

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?

 

[GoCubs12]

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]

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]

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?

Always use radians. 1 deg = π/180 rad. When looking up values for trig functions where degrees is the specified unit of the table or calculator setting change radians to degrees using 1 rad. = 180/π deg. (verify your calculator setting)