Projectile Motion I: Calibrating a Cannon

The "Steps" part is most important, please read it thoroughly.

Prelabs:

a. Relationship between initial velocity and distances.

(Show me your derivation) If you use

,

and get the expression of

,

that's partially correct. Only when

,

then the answer would be correct.

b. Uncertainty propagation.

(Give me the derivation) Since some of you didn't learn partial derivatives, I will give you the detailed derivation (but I won't give you next time).

(1) For the partial derivative of v0 with respect to x, we should treat all but x as constants, thus

.

(2) Similarly, for y, all but y are constants, then we can rewrite the expression as

,

thus

.

Therefore, the uncertainty relationship is

for , but is not acceptable, since means the uncertainty of , which is

,

and that is not equal to

.

Steps:

1. Experiment Design.

State clearly which angle did you choose, how did you measure the distances. For example:

Please use similar format as above in your future reports:

1) Apparatus set up, 2) Data acquisition, 3) Data processing.

2. Measurement and Data.

2.1 Significant Digits

A ruler can measure a quantity up to 0.1 cm, even though you get an integer length in centimeter, append ".0" after the integer value. Thus, for a length of 130.0cm, only the following results are acceptable: (1.300 ± 0.001) m, (130.0 ± 0.1) cm, (1300 ± 1) mm (don't forget the brackets.). The quantity and it's error should have same number of decimal digits of whichever has less (e.g., 5.27 ± 0.021 should be 5.27 ± 0.02, 5.3 ± 1 should be 5 ± 1).

2.2 Rounding (See reference link for more)

2.2.1 Addition and Subtraction -- "to the least accurate place"

Examples:

a. 5.123 + 1.2 = 6.3

b. 1.7 - 10 = -8

2.2.2 Multiplication, Division, Logarithm, Power, etc. -- "least number of significant digits"

Examples:

a. 5.123 × 1.2 = 6.3

b. 1.7 × 10 = 12

c. d.

e.

Note: Do not use letter X as the multiplication sign, you can type it through ALT+0215 (Hold "Alt", Press 0 then 2 then 1 then 5, Then release "Alt") in Windows, and insert × in the character center in Mac.

2.2.3 Constants -- "no effects on others"

When constants are included, they will not affect the rounding of others. Say, 3 + π = 6, 3.1 + π = 6.2, 3.122 + π = 6.264.

3. Uncertainty

Possible sources (systematic/random): continual hitting shifts the rubber mat; the area the ball lands is not a geometric point; not shot at 0˚; etc.

4. Results

See section 2.1 above.

5. Discussion

Come up with solutions! Not just "a more precise apparatus".