Measuring Speed
There are two methods
available which lend themselves to a simple experimental set up.
1. Parabolic Method:
This method involves
firing the projectile at a vertical target some distance, S, from the muzzle of
the coilgun. When the projectile is fired at the target it will fall vertically
by a distance H1. Using this vertical distance and the horizontal distance travelled
we can work out the speed. The diagram below shows the set up.
Fig
1.1
Let's derive an
expression for the speed V. Assuming V is constant then the time taken for the
projectile to reach the target is simply:
Now
we can determine the time taken for the projectile to fall vertically by H1 using:
Now
the initial vertical speed, u, is zero so we can rearrange to give:
Equating
eqn 1.1 and 1.3 gives:
This method requires
that the coilgun accelerator tube is set as close to horizontal as possible otherwise
the calculated speed value will be off  if the tube is pitched up then the speed
will read too high, if it is pitched down then the speed will read too low. The
test rig should be set with a spirit level and the corresponding horizontal projection
point on the target should be set with either a laser beam or a water level 
tube filled with water.
2. Light Gate
Method:
Here we use a pair
of light gates set some distance apart to time the projectile. Using the separation
of the gates and the measured flight time we can easily determine the speed.
Fig
2.1
This
is a much more accurate method of measurement (probably around 5% error) but it
does require some time to build the optical gates and timing system. It is particularly
suited to measuring fast projectiles which would otherwise need a long firing
range to get a significant H1 drop. It is also less sensitive to the pitch of
the accelerator tube. If you intend to use an optical gate system then you will
need an instrument which can measure the time between gate triggering events.
This can be done using a countertimer instrument or a DSO. If you are using a
countertimer then you may need to pay attention to the voltage levels of the
gates when they are clear or blocked.
3.
Ballistic Pendulum:
Conservation
of momentum is the basis of this method. The projectile impacts and becomes embedded
in the pendulum bob which is free to swing about a pivot. After the impact, the
bob swings away and reaches a maximum height which depends on its velocity and
hence energy. By relating the momentum and energy considerations we can arrive
at a projectile velocity based on the pendulum parameters and swing height. Let's
develop an equation for the projectile velocity.
Fig
3.1
All
the motion is considered to be occurring about the pivot point, O. The main assumption
used is that the dimensions of the bob are small compared to the swing length,
L, this allows us to ignore the moment of inertia of the bob about its own mass
centre, in other words the bob is being treated as a point mass. We will also
assume that the mass of the projectile is very small (<x100) compared to the
bob, this means that after the projectile embeds in the bob, the mass centre of
the bob remains virtually unchanged. The projectile is also considered to be a
point mass.
Before
the collision 
The
projectile is travelling at some velocity V_{p}, and directed exactly
at the mass centre of the bob. The bob is stationary. The angular momentum equations
for the projectile and the bob are therefore:
We
can equate the momentum before and after the collision as follows:
Now
the velocity of the projectile and bob are the same after the collision so:
and
therefore we can write:
This
is the velocity of the projectile before the impact. Now we need to determine
the combined velocity after the impact in terms of the swing angle, q.
The
maximum change in height, H_{2}, of the bob/projectile during its swing
gives us the energy of the bob/projectile just after impact. The gravitational
potential energy is then:
Now
H_{2} can be expressed as a function of q as follows:
So
the gravitational energy in terms of q is:
The
kinetic energy of the bob/projectile after impact is:
Rearranging
to give the bob/projectile velocity and substituting in the gravitational energy
expression:
We
can now substitute eqn 3.13 into eqn 3.8 above to give:


Eqn
3.14

If
you are intending to use a ballistic pendulum then you need to remember that this
equation uses assumptions about the physical layout of the system. You need to
keep the swing length long compared to the dimensions of the bob. For example,
a bob which is, say, 100mm in diameter needs a swing length of around 1000mm to
satisfy the point mass assumption. You also need to pay attention to the swing
arm mass, this needs to be light otherwise it will have a significant moment of
inertia about the pivot and will skew the results.
It's
important to realise that the bob mass and swing length need to be large enough
to safely absorb the projectile. Suppose we want a pendulum which can handle a
10g projectile at speeds of up to 100m/s. Let's say we want to keep the swing
angle below 45^{0} and we intend to use a bob of mass 1kg. First thing
we need to do is calculate the projectile/bob energy after the collision, which
in this case is 2.5J. We now use this to calculate the height to which the bob
will swing. This gives us a value of 0.25m. Since we are looking to keep the maximum
swing angle below 45^{0} (for safety considerations), we need a swing
length of 0.85m. This is quite large and reflects the substantial energy which
a fast projectile carries. You'll need to decide on the best setup for your particular
situation. Most low voltage (~50V) single stage coilguns will be lucky to get
beyond 25m/s so a smaller pendulum is feasible.
There
are two possible ways of measuring the swing angle  either attach a pen to the
top of swing arm and let it trace the swing on a piece of card, or, attach a potentiometer
to the arm pivot and record the voltage output, this can then be related to the
angle.
