Projectile Motion Lab: Accuracy and Factors Affecting Range

[lilyxmo]

Gr. 11 Projectile motion lab?

Homework Statement

we are conducting an experiment where a ball is being shot from a cannon from a height of 1.435m
we have already calculated our initial velocity, which is 5.442m/s, by timing and measuring when the ball is shot and lands 5 times. Then find the velocity of those and find the average..

BUT we have to predict where our ball will land and be able to hit a target accurately if given an angle.

2. Homework Equations and attempts at solving
SO i tried calculating it if we shot the ball at an angle of 35 degrees
and i used this formula:

Δdy = vyΔt + 1/2gΔt^2

then set [up] as positive
so.. gravity is -9.8m/s^2
dy = -1.435m
and vy = 3.121m/s (found the components using vy = v1sinθ)

so i solved for t and got t=0.946s

theeen subbed in (t=0.946) in
dx = vxΔt
where vx = 4.458m/s (solved with v1cosθ)

and got the range of 4.22m



BUT when we actually tried shooting the ball, it landed about 3.25m away which is VERY off from my calculations

We even tried shooting it at 45 degrees to maximize the range and still got a distance of 3.25m! Our distance at 20 degrees was 3.5m, which is greater than the "maximium range"

i really don't understand how to accurately find the range because all my calculations are all off. Is there another factor that needs to be considered? Apparently air resistance does not matter because the ball is only in the air for such a short period of time..

 

[CWatters]

Is the launch velocity dependant on the launch angle (perhaps due to the way the cannon works?)?

 

[BruceW]

hi lilyxmo, welcome to physicsforums :)
Ah, your method looks correct to me. For the angle of 35 degrees, I get the same answer for the horizontal distance traveled. As CWatters says, maybe your actual cannon does not exactly hold true to the underlying assumptions of your theoretical model.

Also, since your cannon is not at ground level, shooting at 45 degrees will not give the maximum horizontal distance. (This is in the simple theoretical model). If your cannon was at ground level, then 45 degrees would give the maximum horizontal distance. But that is not the case here. You can work out what angle gives the max horizontal distance, for a given height of the cannon. But since your actual cannon is giving strange results, this might not be true for the actual cannon.

 

[SteamKing]

Like BruceW said, perhaps the underlying assumptions of your theoretical model require re-examination.

The one factor which you have presumptively dismissed is the effect of air resistance on your range. In even a simple drag model, drag due to air resistance is proportional to the velocity squared.

Assuming your projectile is spherical and smooth, the drag coefficient Cd = 0.5

The drag force D = 0.5 Cd (ρ/2)Av^2,

where v is the velocity
ρ is the density of air (approx. 1.2 kg/m^3)
A is the frontal area of the sphere (πr^2)

Considering drag is never a simple proposition, but I think it may help explain the discrepancy in your range values. As a first cut, I would try to estimate how much range reduction you would experience due to drag by considering the effect only on the horizontal velocity of the projectile.

 

[haruspex]

How did you measure the velocity so accurately?
The three range figures you give, same at 45 and 35 but more at 20, is contrary to what one would expect. It suggests some degree of experimental error. How reproducible were these results?

 

[barryj]

What is your ball made of? A tennis ball would suffer more from drag than an iron ball given the sme angle and initial velocity conditions.

 

[verty]

I haven't checked the calculations but I think the initial velocity will vary at different angles. Because the chamber behind the cannonball is not airtight, most of the acceleration will happen immediately after firing. When the cannonball is fired horizontally, it'll have a certain acceleration, but firing it vertically, it'll accelerate more slowly (the weight is subtracted) and due to leakage of air, the pressure will be less versus distance. This must surely mean that the cannonball will have a lower velocity.

So there will be variation with angle. Perhaps it is enough to account for the errors you are seeing. What I can say is, it would cause the maximum range to happen at a lower angle than 45 degrees. An interesting experiment would be to find the angle of maximum range with your cannon. That would give an indication of how strong this variation effect is.

Coincidentally, even if the chamber is airtight, there will be variation, but with leaking air the variation will be increased.

 

[lilyxmo]

This is an experiment for school so we can't use real cannons. So we just used a marble and this "cannon" that's made of a pipe and a spring, where we push the spring down and lock it in place with this needle, then release to shoot the marble. I also counted how many spring 'loops?' that i push down each time so the same force is applied each time.

I measured the velocity by shooting the 'cannon' at 0 degrees 5 times from a height of 0.75m and measuring the distance where it fell and also timing it. Then using the formula, dx = vxt, i found vx for all my testing, and got the average of the 5, which is the initial velocity?

 

[lilyxmo]

thank you BruceW for the welcome :)

so i found drag force to be 7.037N if the marble has a diameter of 15.875mm? So the surface area of the marble is around 0.792m^2
How do i use this? @.@

 

[BruceW]

I measured the velocity by shooting the 'cannon' at 0 degrees 5 times from a height of 0.75m and measuring the distance where it fell and also timing it. Then using the formula, dx = vxt, i found vx for all my testing, and got the average of the 5, which is the initial velocity?

So your average vx for the horizontal cannon turned out to be 5.442 m/s ? Yes, that should work. Also, as haruspex mentioned, the initial velocity might vary quite a bit each time you shoot the cannon. For example, you got a range of 3.25m but you expected 4.2m theoretically. Now, if you repeat the experiment and get ranges of 3.25m , 5.1m , 4.5m , 3.6m Then you can imagine that your experiment maybe does agree with the theoretical result, but due to experimental uncertainty, you get a different answer each time.

But if you are consistently getting a range which is close to 3.25m, then yeah, it might be either due to air resistance or perhaps the marble moves through the cannon differently when the cannon is tilted, so the initial velocity actually has some dependence on the angle of the cannon as CWatters mentioned.

 

[BruceW]

thank you BruceW for the welcome :)

so i found drag force to be 7.037N if the marble has a diameter of 15.875mm? So the surface area of the marble is around 0.792m^2
How do i use this? @.@

no problem! Uh, wow, you measured a drag force? Usually the drag force depends on the velocity of the marble. But I guess you measured the drag force for a 'typical' velocity? Well, before your model was assuming the only force was gravity. So now you have a value for the drag force, you could also take that into account in your model.

edit: The equations will be a bit more complicated when you introduce a drag force. Maybe it is best to do as SteamKing suggested, and only apply the drag force to the horizontal direction. This is an approximation, since the 'true' drag will also oppose vertical velocity, but since the vertical velocity will be less (I'm guessing), then most of the velocity will be in the x direction for most of the time, so maybe it is an OK approximation. Also, is it really 7.037N drag force? this seems like a lot for a marble.

 

[Javier21]

Did you count range as rolling range or where it originally landed? It makes a big difference. It should be where it originally landed.