What Happens to Projectile Speed When Mass Increases Mid-Flight?

[Strasz]

While playing a rousing game of Dungeons and Dragons, we came upon a strange question that I thought might best be answered by the fine people at these forums.

A tree polymorphed into an arrow, when fired from the bow will transform back into a sharpened log. What then happens to the speed of the projectile? Will it continue flying at that speed? Or will it slow down? Will it simply fall to the ground?

Basically, what happens when an object in motion suddenly, dramatically increases in mass? Assume standard gravity and air resistance.

 

[cepheid]

Objects don't suddenly increase in mass, but if they did, momentum would still have to be conserved. Let primed quantities (with a prime symbol ' after them) represent the quantities after the "mass increase." Momentum is mass x velocity, so

mv = m'v'

v' = m/m' v

The velocity would decrease by a factor m'/m.

Edit: you asked about gravity too. Since the horizontal velocity is reduced, but the vertical accleration remains the same, the vertical velocity increases at the same rate as before the mass increase, and so the time taken to reach the ground is the same. Therefore, the object will not travel as far horizontally before reaching the ground as it would have if the mass increase hadn't occured. Another way to think about it: the shape of the parabolic trajectory of the object would change.

 

[Strasz]

Thanks a bunch!

 

[torquil]

It depends on what "polymorphing" is. Does it violate or conserve momentum...

If we are breaking the laws of nature anyway, we might as well break the conservation of momentum.

Look at it from the arrows perspective, i.e. you are an observer sitting on the arrow. Why should the world around the arrow suddenly slow down just because the mass of the arrow increases? That seems even worse to me than violation of conservation of momentum.

 

[PJC01]

I must first clarify that the scenario described is not possible in our current understanding of physics. The concept of a tree polymorphing into an arrow and then transforming back into a log is not supported by any known scientific principles.

However, for the sake of exploring the question, let us assume that this scenario did occur. In this case, the speed of the projectile would change based on the conservation of momentum and energy. The mass of the object has suddenly increased, which means that the momentum must also increase in order to conserve the overall momentum of the system.

Therefore, the speed of the projectile would decrease as it transforms from an arrow to a log. This is because the same amount of energy is now being distributed over a larger mass, resulting in a decrease in velocity. Additionally, the increased mass would also result in an increase in air resistance, further slowing down the projectile.

In terms of gravity, it would depend on the specific circumstances of the transformation. If the transformation occurred while the projectile was still in the air, the increased mass would cause it to fall to the ground faster. However, if the transformation occurred after the projectile had already reached its peak height, the increased mass would not have much of an effect on its trajectory.

Overall, the speed of the projectile would slow down due to the increase in mass, and it would also be affected by air resistance and gravity. It is important to note that this scenario is purely hypothetical and does not align with our current understanding of physics.