Optimizing Projectile Range: The Physics of Mass and Launch Velocity

[houlahound]

Not the best terminology but I have in mind impedance matching as an analogy to this question.

Throw a rock by hand or launch a projectile with a device. Start with very low mass projectiles through to so massive the device can not generate enough power to launch.

There is a certain mass projectile that will give the greatest range so range versus mass will have a peak. The graph will be different for every launcher.

Easy to test this with throwing stuff of different masses, same drag.

What is the physics here?

 

[A.T.]

What is the physics here?

For a device that does the same work on the projectile, regardless of mass:
- Projectile density -> ∞ : The launch speed goes to zero
- Projectile density -> 0 : Drag stops the projectile immediately

 

[Baluncore]

You are transferring stored energy in the launcher into kinetic energy in the projectile. The problem is that as you release the projectile, if part of the launcher is still moving, then some KE has not been passed to the projectile. There are ways of getting almost complete energy transfer. One way is to match the momentum and the kinetic energy of the launcher and projectile.
Newtons cradle is a demonstration of that matching. https://en.wikipedia.org/wiki/Newton's_cradle

Maximum power is transferred in an electrical circuit when the impedance is matched.

A whip is a tapered impedance transmission line in which the energy of a high mass at low velocity is transformed into a low mass at high velocity. You might consider placing a matched projectile on the end of a virtual whip.

 

[tech99]

I believe that a generator supplies only half its power to a load, the remainder being dissipated in the source impedance. So max efficiency is 50% under matched conditions. On the other hand, better efficiency can be obtained by mis-matching, but the power in the load is less.

 

[Baluncore]

I believe that a generator supplies only half its power to a load, the remainder being dissipated in the source impedance. So max efficiency is 50% under matched conditions.

I think you are confusing the matched impedance of a transmission line with the series resistance of a power source driving a resistive load.

An impedance matched junction between two transmission lines transfers 100% of the energy without reflection. The line impedance is a function of line inductance and capacitance. Neither inductance nor capacitance dissipates real power. The series resistance along a transmission line wastes real power. That is why high voltages and thick wires are used to distribute power. It is to keep the source resistance low.

It is true that to get the maximum of 50% of the available power transferred, the resistance of a load must be matched to the series resistance of a source. But we are not dissipating energy by friction in the launcher, nor resistively heating the projectile, we are giving it kinetic energy efficiently. The game will not be over until projectile impact converts the KE to heat.
By continuing to oscillate for some time, Newton's cradle demonstrates that better than 95% energy transfer is possible in a projectile launcher.

A launcher is able to transfer 99% of the available energy to the projectile. That is because there can be very low frictional losses in the launcher energy transfer mechanism.

 

[houlahound]

Can we get this mathematical, say the simplest possible system. You have a stick (baseball bat) of fixed physical properties. You have spheres of fixed elasticity and volume but different masses.

How would you choose in advance of any trial which sphere would have the most range given a fixed launch angle and launch velocity.

Or instead use a spring launcher inside a launch barrel if that is simpler?

 

[Baluncore]

Or instead use a spring launcher inside a launch barrel if that is simpler?

Consider; 1. A light weight spring mounted against a solid wall, with a heavy ball attached at the free end. 2. Place another identical ball against the spring end ball. 3. Compress the spring with the attached ball. 4. When released, the spring will accelerate the ball until the point of ball impact. 5. The spring-ball will then remain where it was, the free ball will be launched. At that moment you have a launcher that satisfies the available mathematics of Newton's cradle. Everything is acting on a straight line so 2D and 3D vectors can be avoided.

You might consider modelling a golf club hitting a golf ball. The shaft weighs very little, but the club head is heavier than the ball. Again, the result of the collision will be based on the conservation of energy and momentum. The centroid of the club head travels along an arc. That arc passes through the ball centre. Study the ball velocity for different mass ratios of club head to ball.

How complex do you want to make it ?
When you swing a baseball bat you are transferring low velocity muscle energy into the ball velocity via an impedance matching device called a bat.
Assume everything is elastic and so is lossless. Assume the mass of the ball and bat are known parameters. The geometrical linkage of bones and muscles accelerates and steers the bat in a complex way, (you will need to simplify that). The bat with mass and velocity vectors, contacts the moving ball, also with mass and velocity vectors, to deflect the ball onto a different vector. Conservation of energy and momentum in the 3D system dictate the initial and final velocities of the ball and bat.
Impact on the “sweet spot” or centroid of the bat makes the math easier as it eliminates bat rotation inertia from the model. Would you expand your model to handle a glancing blow of bat against ball?