# Why is efficiency constant?

1. Aug 14, 2010

### donkeycopter

I need to explain why, in our tests, the efficiency of our slingshot device remained constant when the angle of launch was changed and the extension (how far the rubber band was pulled back) was constant.

Basically it was a 1m long wooden plank held up on bricks, with a rubber band attached to 2 nails at the top of the board. We put a rubber ball in the rubber band and shot it from different angles.

We calculate efficiency in terms of energy lost.

Thanks!

2. Aug 14, 2010

### 6Stang7

Well, what caused the energy loss?

3. Aug 14, 2010

### donkeycopter

See I've identified air resistance and friction between the ball and the plank as how the ball loses energy, but I don't really know. Why would that not be effected when the angle changes?

4. Aug 14, 2010

### donkeycopter

shouldn't the weight vector down the plane effect the efficiency when angle is changed? Why does it not?

5. Aug 14, 2010

### hikaru1221

Can you provide some more details, such as the weight of the ball (or how big it is, what material it is made of), the dimensions of the rubber band?

6. Aug 14, 2010

### rock.freak667

$$\eta= \frac{Energy \ Output}{Energy \ Input}$$

What is type of energy is the input (rubber band)?
Since the mass lands on the ground, the final energy at the ground is kinetic.

7. Aug 15, 2010

### donkeycopter

The weight of the ball is 0.015g. It is a rubber bouncy ball. The rubber bands is about 15cm on each side.

Why doesn't the angle affect the efficiency?

8. Aug 15, 2010

### hikaru1221

I think it does actually. If we consider the output energy is kinetic energy, then there are 2 factors making the ball lose its energy: friction and gravity (strictly speaking, there are 3, the other of which is air resistance; but it's trivial in this experiment).
However the output energy is nearly the same even though the angle changes. Here is the reason:
- The Young modulus of rubber is about E = 107 Pa. Suppose that the rubber band has width of 1cm, thickness of 1mm, then the elastic constant of this rubber band is about k=1333 N/m. If we stretch the rubber band from 15cm to 20cm (let's hope that Hooke's law still works!), then the elastic energy, or input energy, is Ei = 1.67 J.
- The ball goes through a distance of d=5cm (from 20cm to 15cm) before it leaves the rubber band. In the extreme case, where the plank is raised vertically, the energy lost to gravitational energy is Eg = mgd = 0.0075 J (I guess you meant m=0.015 kg or 15g).
- Friction on the ball F = rN, where r is the frictional coefficient and N is the normal force. In the extreme case, where the plank is set horizontally, N=mg. Suppose that r=1 (r=1 is very rare and very large too!). Therefore: F=0.15 N. The work done by friction when the ball leaves the rubber band: W = Fd = 0.0075 J.
You can see that W and Eg << Ei. That means, the energy lost is infinitesimal compared to the input energy, so output energy nearly equals to input energy. The efficiency is nearly 100%.

Last edited: Aug 15, 2010
9. Aug 15, 2010

### donkeycopter

Thanks very much Hikaru!

The thing is, efficiency is about 40% in reality.

Why is air resistance trivial? I was under the impression that was the main component slowing the ball down?

I think you may have calculated the coefficient incorrectly, its closer to 200Nm that 1333.

But, as my tests show, the angle doesn't effect efficiency. Can anyone supply a good reason for this happening? I don't need explanations of what should happen - but why whats happened occurs!

Thanks! :)

10. Aug 15, 2010

### Gyro

I think you're right about it being the main component. But at what angle does air resistance act relative to projectile's velocity?

EDIT: I think this should have been my first response to you. Sorry if so.

Last edited: Aug 15, 2010
11. Aug 15, 2010

### rock.freak667

Last edited by a moderator: Apr 25, 2017
12. Aug 16, 2010

### donkeycopter

but why does the angle not effect efficiency?

@rock.freak667: How is that relevant?

13. Aug 16, 2010

### hikaru1221

Then how do you calculate efficiency anyway? I doubt it's not the slingshot's efficiency.
About the elastic coefficient, I use the standard value of rubber, which can be found on most websites. The material used to make the band is one factor. Another factor is the extension length. Hookes' law only applies to rubber in small range.

14. Aug 16, 2010

### donkeycopter

I calculated efficiency as the real KE over the ideal KE.

But thats irrelevant, my teacher has confirmed my efficiencies are accurate!

What I need to know is a reason the angle doesn't effect the efficiency? Just a reason to describe this (if it is) anomaly?

Thanks!

15. Aug 16, 2010

### hikaru1221

How do you calculate the real KE?

16. Aug 16, 2010

### donkeycopter

using KE = 1/2mv^2

I find v by working backwards from my measured ranges.

If you aren't sure why the efficiency is constant, this discussion would seem pointless?

17. Aug 16, 2010

### rock.freak667

Because in that question, your formula for efficiency did not include the angle, it is a similar type of exercise.

18. Aug 16, 2010

### Gyro

A leads to B leads to C ... Before you can get to why (or why not), answer my question first.

At what angle does air resistance act relative to the projectile's velocity? And at what angle does the projectile's velocity act relative to the launch angle? Are these angles constant, do they change, how so? When you change the launch angle, how do the other angles change? What does changing the launch angle do the time the projectile is in the air? How does that affect the amount of air resistance that acts on the projectile?
Go from there.

Last edited: Aug 16, 2010
19. Aug 18, 2010

### whymustzbelas

i am doing this experiment right now - literally

even though your teacher said it was correct, i don't think your way of calculating efficiency is the best. efficiency is the energy lost:

efficiency = (KE) / (EPE + GPE)

also
the angle does affect the efficiency. due to the different components, the angle affects the initial velocity and thus, the range. so the efficiency will change depending on the angle