How to maintain the same initial velocity in trials with projectile motion experiment?

Question

If I would like to experiment with projectile motion to determine the relationship of launch angle and horizontal range (for instance, I would throw a ball multiple times from a height with varying angles), how could I make sure that every trial has the same initial velocity?

If similar initial velocity could not be obtained in every trial, is it suggested to include this in the limitation of the study or should I just discuss that this is one source of error in determining the relationship of launch angle and range?

Answer 1

Perhaps the easiest and the most inexpensive way I can think of is to bounce the projectile -- say, a ball -- off an inclined ramp. The height from which the ball is released determines the speed of the ball after it bounces (the initial velocity for the rest of your experiment), whereas the angle of the incline controls the launch angle.

Note that this procedure assumes the frictional loss of energy as the ball moves through the air before hitting the ramp, as well as the energy lost to heat and sound as it bounces off, remains consistent throughout your trials, which is reasonable.

You could also use a ball-shooter like they utilize in racquet-and-ball games such as tennis, but procuring that might be difficult.

Answer 2

If I understand your question correct you want to reach something like in this plot?.

you start to throw the ball from different heights , you want th reach the same distance , holding the start velocity constant and changing the shouting angle.

if so here is the solution:

the trajectory of the ball is :

$$y={\frac {\sin \left( \varphi \right) x}{\cos \left( \varphi \right) } }-\frac 12\,{\frac {g{x}^{2}}{{v_{{0}}}^{2} \left( \cos \left( \varphi \right) \right) ^{2}}}+y_{{0}} \tag 1$$

where $~y_0~$ is the start height

substitute into Eq. (1) the required range $~x=x_r$ , from here you can obtain with $~y(x_r)=0~$ the required angle $~\varphi_r~ $ holding the start velocity constant.

$$y(x_r)=0={\frac {\sin \left( \varphi _{{r}} \right) x_{{r}}}{\cos \left( \varphi _{{r}} \right) }}-\frac 12\,{\frac {g{x_{{r}}}^{2}}{{v_{{0}}}^{2} \left( \cos \left( \varphi _{{r}} \right) \right) ^{2}}}+y_{{0}} \tag 2$$

from the solution of Eq. (2) you obtain $~\varphi_r=\varphi_r(x_r,v_0,y_0)$

Edit

you can also get this results.

you start with different heights , all of the projectiles reach the same height and the same distance, but you have to verify the initial velocity and the initial angle .