Projectile Motion II: Hitting a Target

Pre-lab Hints:

(Show me the derivation and final answer in your report):

1. For a ball landing within a range of (-x0, x0) under normal/Gaussian distribution with mean µ and standard deviation σ, the corresponding probability is,

. (i)

2.

(ii)

3.

(iii)

4.

(iv)

Lab:

Your calculated results should be accompanied by the equations which yields the results! Otherwise people won't know how did you get them.

Do not copy my equations!

There're 3 occasions in this lab: maximum range (1), horizontal displacement (2), and horizontal & vertical displacement (3).

0. Calculating the probability:

Using Eq. (i), replace x₀ with the radius of the basket, σ by the uncertainty of the x-distance, and µ by 0. Since normal distribution is not integratable, human provides the cumulative distribution function (CDF) for it, which is often represented by

.

And the corresponding probability from -x₀ to x₀, when µ = 0, is

.

You can get the CDF value from here.

1. While calculating the standard error for distance in occasions (1) and (2), equation (iii) above is correct, but there's one important thing you should keep in mind, that is: you should not plug in to equation (iii) directly! You should convert it to radian, which is

rad, and plug it into eq (iii)! All of you lose points here.

2. Do not use eq (ii) to get the angle for a specific distance -- you will get only one result! You should use eq (iv) above.

3. While comparing which angle is more reliable in occasions (2), you should plug the two angles into eq (iii) and compare the results.

4. Eq (iii) cannot be applied to occasion (3), since this equation is only valid for horizontal displacement (but no one pointed that out in your lab reports)! You can check the 3rd question in the prelab for details. Here are the specific steps and equations:

x-direction:

, (4.1)

y-direction:

. (4.2)

From Eq. (4.2), we can get the time the cannon ball travels when it hits the basket, which is

, propagated from velocity and angle, represented by and (we will treat y as fixed value, since the basket will only measure whether the cannon ball falls into it in the x-direction), which is very complicated.

Using Eq. (4.1), we get the uncertainty of x by,

. (4.4)

Plugging the representation of into Eq. (4.4), you will get the error in x-direction. Then you can calculate the probability using step 0 above.