Question

Data Analysis (10pts) Fill in the following data table: Setup Davg (cm) SD (cm) D ΔΕ/Ε theo (cm) 1 2 3 Conclusion (4pts total, 1pt each) 1) For each trial, compute the agreement between D and Dtheo - Do they agree? 2) Did the ball launch with an initial vertical velocity? How do you know? 3) Identify a source of systematic error present in the experiment. 4) How would you improve the experiment to reduce the effects of that source?
Please help answer the questions. Thank you!

Answer 1

For the given experiment,
Launch height is L
Range is D
then
time of flight be t
intial velocity be v and angle of projectile be theta
then
D = v*cos(theta)*t
and
-L = v*sin(theta)*t - 0.5gt^2
0.5gt^2 - v*sin(theta)Tt - L = 0
hence,
t = (v*sin(theta) + sqrt(v^2*sin^2(theta) + 2gL))/g
so,
D = v*cos(theta)*(v*sin(theta) + sqrt(v^2*sin^2(theta) + 2gL))/g
D = v^2*sin(2*theta)/2g + v*cos(theta)*sqrt(v^2*sin^2(theta) + 2gL)/g
so, we can fing Dtheo from the above formula as theta is given for each trial, L = 0.78 m for all trials,
but we dont know v
so we use
2*g*h = v^2*sin^2(theta)
v^2 = 2gh/sin^2(theta)
h is given for each trial

so the completed data tables are as under

Distance Measurements
Setup	Angle (deg)	h (cm)	D1 (cm)	D2 (cm)	D3 (cm)	D4 (cm)	D5 (cm)	Davg (cm)	del(D) (cm)
1	54	31.2	91.8	92.1	93	95.5	96.2	93.72	1.797109
2	62	40.8	107	107.5	107.7	108.8	113.4	108.88	2.335294
3	72	52.4	122.2	123.5	124	125.5	125.7	124.18	1.301384

Setup	Davg (cm)	del(D) cm	Dtheo (cm)	del(E)/E	v^2
1	93.72	1.797109	1.301526		9.352728
2	113.4	2.335294	1.174234		10.26808
3	125.7	1.301384	0.877684		11.36626

Conclusions
1) D, Dtheo have been computed, they are realy close

2) We know the ball launched with initial vertical velocity as the ball travels some height h up the launch height of 78 cm everytime,
this means the ball travels up, which means it needs to have initial upwards vertical velocity
  
3) We are measuring distance and we have less time to do this as trajectory of the ball keeps on changing every moment
so there is a scope of error in measuring distances this way
also, since we are ignoring air resistance in our theoretical calculations, but the effects of air resistance do creep up inthe results hence we have to take care of that otherwise our error will be high
  
4) To reduce3 the effects of air, one can try and perform the experiment in a vaccum sealed room using automated launch sequence (like using electromagnet switch to release the ball) etc