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_it+(1/2)at²

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_iCosQ V_vertical= V_iSinQ

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

t₁= (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_it+(1/2)at²

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_it+(1/2)at²

After some algebra determine

t₂=(-v_i+[v_i+2ad]^1/2)/a

noting that t₂ 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=t₁+t₂

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!!