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
.
You can use
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:
Cannon number: #99
1) Put a carbon paper under a white paper, and when the ball hits the paper, it will record the landing point.
2) Shoot a cannon ball with an angle of 0°, use a metal sinker to get the horizontal position where the cannon ball starts moving, measure the distance between the initial horizontal position and the landing point using a ruler, and get the horizontal distance the ball travels. Use a ruler to measure the vertical distance the ball travels, which is from the muzzle to the point where the sinker lands.
3) Repeat 25 trials, put the data into table and calculate the mean using AVERAGE function in Excel, standard deviation (s.d.) using STDEV.S in Excel and standard error (s.e.) from
.
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".