The Physics of the Project:

To understand projectile motion, we must understand that all projectiles travel in a parabolic flight path. Then we need the following formulas to understand this particular system:

V=d/t V_{final}=V_{i}+at d=V_{i}t+(1/2)at^{2}

Using these three
formulas we can describe the entire system.
Let us start by giving the projectile some initial velocity. In our setup, 2 photogates allow us to
obtain a time for which the swing arm takes to travel a certain known distance
d. We can therefore use V=d/t to
calculate the velocity of the projectile as it leaves the swing arm. Call this velocity V_{i}. Then we must break down this velocity into
it’s component velocities. In other
words, we must find our horizontal and vertical components, since the
projectile is launched at an angle, and we call this angle Q,
and it is taken in relation to the horizontal.
To do this use

V_{horizontal}=V_{i}CosQ V_{vertical}= V_{i}SinQ

Now that we have our
component velocities it is necessary to understand that horizontal velocity
will be unchanged during flight, that is to say it has no forces acting on it
if we neglect air friction. Only the
vertical velocity is accelerated by gravity.
So the first step is to find the time it takes for the projectile to
reach maximum height. Use:

V_{final}=V_{i}+at

Through algebra
determine that

t_{1}=
(V_{final}-V_{i})/a

remembering that our
acceleration is negative due to the motion of the projectile opposing
gravity. Note the initial velocity is
initial vertical velocity, and that final velocity is 0 since the projectile
has no velocity in the vertical direction when at the top of it’s flight path.

We can now determine
the distance traveled upwards using:

d=V_{i}t+(1/2)at^{2}

again making sure we
note acceleration in this case to be negative.
Note well that this is the distance traveled upwards from the swing arm,
NOT the distance from the ground. To
get total max height, add the length of the swing arm. Now we have a total distance to the
ground. Knowing this we can determine
time it takes the projectile to fall to the ground, again noting that it is not
simply the time it took the projectile to reach maximum height. DO NOT make that assumption. Again using

d=V_{i}t+(1/2)at^{2}

After some algebra
determine

t_{2}=(-v_{i}+[v_{i}+2ad]^{1/2})/a

noting that t_{2}
is time the projectile takes to fall to the ground.

Now we have a total
flight time, again noting only the vertical velocity affects such a
calculation. So total fight time:

t_{total}=t_{1}+t_{2}

The only thing left to
do is calculate the range of our system.
To do so, we need to use the horizontal velocity and the relation

V=d/t

So using this relation
and after a little algebra

(V_{horizontal})(t_{total})=d

and in this case d is
our range or the maximum horizontal distance our projectile will travel.

All problems that
involve projectiles can be solved using the three aforementioned simple
formulas and this basic breakdown. This
should be solid in your mind as, on exams questions worth a lot of marks are
based on this same problem year after year.
Learn it well, and follow my steps, and you’re sure to reap in the
marks!!