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Projectile Energy Example

This page is to aid in energy analysis when shooting/throwing objects. 
Written by Daniel Yang ( SP18

Part 1: Calculating the Kinetic Energy of the Projectile:

The first step is to calculate the velocity of the projectile. If launching the ball horizontally, you can easily calculate the velocity of the ball (distance/travel time). If you need a projectile to travel a certain distance and height, you should use projectile motion (Link)

Keep in mind in the real world, air resistance (drag) significantly hinders the distance and speed of light weight projectiles. For example, you can throw a golf ball much further than similar sized pingpong ball. This is because the golf ball has a lot more mass (and momentum) to overcome drag. To compensate for drag you should consider using a large design factor of safety.

Once the initial velocity is found, energy can easily be found with the kinetic energy equation:
E_req = .5mV^2

Part 2: Tosser Example - Calculating the Power Required:
In this example we will consider a spring loaded catapult. We model the catapult as a projectile on a cart being pulled by a spring. 
Vo = final launch velocity (determined in part 1) 
Δx = distance the spring pulls the cart. Δx can be the maximum length of the spring.
m = mass of ball and cart

In order to calculate the power needed, we need to estimate the amount of time the spring will be applying energy to the ball. In order to do this, we will do a "back of the envelope" calculation of the contact time (note: actual envelope is not required).

We will estimate the contact time by assuming that the projectile travels at Vo for entire distance Δx.

Time_contact = Δx/Vo
Calculating power we obtain:

Power_req = E_req/Time_contact
Discussion of results:
Please note that this is a very rudimentary power calculation. Time_contact will be faster in reality because it will take time to accelerate the cart up to Vo. However, this calculation should give us a conservative estimate of the power needs of launching the ball.

Part 3: Pitching Wheel Example - Calculating the Power Required:
In this example we will consider a pitching machine. A ball is fed through two counter rotating flywheels. Kinetic energy of the flywheels is transferred to the ball.
m = mass of the ball
r = radius of flywheel
ω = speed of flywheel (rad/s)
Vo = speed of the ball

For simplicity of this calculation, we will assume that both wheels are powered by only one motor (this can be done with three gears).

First we must calculate the speed the flywheels need to spin. Assuming no slip between the flywheel and the ball, ω is given by:
ω = Vo/r

Next we need to calculate the rotational inertia of the flywheel:
I = .5*m*r^2

Now that we have ω and I we can calculate the kinetic energy of the flywheels. We multiply by 2 because there are two flywheels.
KE_flywheel = 2*(.5*I*ω^2)

We will now use power to calculate a theoretical flywheel spin up time. 
Time_spinup= KE_flywheel/Power_available 

Discussion of results:
If we assume that the flywheels lose all of their energy when contacting the ball, Time_spinup can give us an estimate of the maximum rate of fire. Also, note that we did not calculate the power available to launch the ball. This is because it is too hard to model the exact contact of the ball with the flywheels. Also, the power available in a flywheel design is only dependent on KE_flywheel and how fast you can transfer KE_flywheel to the ball.