Conservation of Linear Momentum
The recoil force or kick from a
launcher can be calculated using the principle of conservation of
linear momentum. The recoil force you will experience while firing a
launcher depends on the mass of the projectile, mass of the launcher, and
velocity that the projectile leaves the barrel.
The linear momentum of a rigid
object in this case is equal to the mass of the object multiplied by its
velocity. If we set the sum of the system's linear momentums equal to the
other system's sum of linear momentums then we have conservation of linear
momentum as shown:
| (Σ Syst. linear momentum)1 | = | (Σ Syst. linear momentum)2 |
As shown in the figure below,
the projectile and launcher have been considered as a single system, since
the impulsive forces, F, between the launcher and projectile are internal to
the system and will therefore cancel from the analysis. So during the time
it takes for the projectile to leave the barrel, Δt (about 0.03 seconds),
the handles which are attached each exerts a nonimpulsive force Fs on the
launcher. This is because Δt is very short, so that during this time the
launcher only moves through a very small distance. As a result, Fs = ks 
0,
where k is the stiffness of the launchers support (much like a spring
stiffness).
0,
where k is the stiffness of the launchers support (much like a spring
stiffness).
Schematic of projectile being fired (Reproduced without the permission of Engineering Mechanics and Dynamics, R.C. Pearson Prentice Hall, Upper Saddle River, NJ)
From this it is shown
that momentum for the system is conserved in the horizontal direction.
In order to find out the
recoil force, we first need to find what speed the launcher is recoiling at.
Using the diagram above, we can write the equation for conservation of
linear momentum for the launcher:
where
| mc |
=
|
Mass of cannon |
=
|
weight of launcher divided by earth's gravity acceleration | |||
| mp | = |
|
|||||
| vc 2 | = | Velocity of cannon/launcher | |||||
| vp 2 | = | Velocity or speed that the projectile leaves the barrel. This can be estimated but later verified with a chronograph or similar device | |||||
So this shows that the
launcher is moving back at a speed of 5.2 feet per second.
The average impulsive force
exerted by the launcher on the projectile can be determined by applying the
principle of linear impulse and momentum to the projectile (or to the
launcher). Using the principles
of linear impulse for a rigid body we can write,
where
Therefore we now plug in our values
Now we solve for the recoil
force
where
| Favg |
=
|
Recoil force | |
| Δt | = |
|
NOTE: Using these equations you can try different projectile weights
and see how it affects the recoil of your launcher.
Tidak ada komentar:
Posting Komentar