Projectile Motion Problem: rocket-propelled arrows

[Fenrir]

First of all, I apologize if this type of question is in the wrong category

Question: The Wubei zhi was a military treatise written in China in 1628. Inside its 80 volumes aredescriptions of many pyrotechnic devices. A rocket-powered arrow (firework) is described ashaving a range of 400 paces. An historian rings you up and asks you to estimate the maximumspeed of these rockets. Applying the usual treatment of projectiles, what would your estimatebe?

I don't really know where to start as there is no angle of launch (maybe 45 degrees?) or other data, apart from the distance being 400 paces (which is about 300m?)

Any help to push me in the right direction would be greatly appreciated!

Many Thanks

 

[Deleted member 4993]

First of all, I apologize if this type of question is in the wrong category

Question: The Wubei zhi was a military treatise written in China in 1628. Inside its 80 volumes aredescriptions of many pyrotechnic devices. A rocket-powered arrow (firework) is described ashaving a range of 400 paces. An historian rings you up and asks you to estimate the maximum speed of these rockets. Applying the usual treatment of projectiles, what would your estimatebe?

I don't really know where to start as there is no angle of launch (maybe 45 degrees?) or other data, apart from the distance being 400 paces (which is about 300m?)

Any help to push me in the right direction would be greatly appreciated!

Many Thanks

Did you want the maximum speed or the minimum speed of the rocket (to have a range of 300 m)?

 

[Ishuda]

First of all, I apologize if this type of question is in the wrong category

Question: The Wubei zhi was a military treatise written in China in 1628. Inside its 80 volumes aredescriptions of many pyrotechnic devices. A rocket-powered arrow (firework) is described ashaving a range of 400 paces. An historian rings you up and asks you to estimate the maximumspeed of these rockets. Applying the usual treatment of projectiles, what would your estimatebe?

I don't really know where to start as there is no angle of launch (maybe 45 degrees?) or other data, apart from the distance being 400 paces (which is about 300m?)

Any help to push me in the right direction would be greatly appreciated!

Many Thanks

If we use the usual ballistic equation with a convenient co-ordinate system, we would have that the arrow is at ground zero at -200 paces and at +200 paces with a height h at 0 paces giving the vertical distance (y) equation
y(x) = h - x²
where x is the horizontal distance. Since y=0 at \(\displaystyle \pm\)200 paces
h = 200²
Now, what have you been taught about the relationship between velocity and distance?

Note that the equation gives you the velocity along the path of the arrow and not necessarily in the x or y direction.

EDIT: Actually the equation should contain an acceleration term [y(x) = h - a x²] which I ignored since otherwise there isn't enough information for a solution (as implied by S Khan).

 

[stapel]

First of all, I apologize if this type of question is in the wrong category

Question: The Wubei zhi was a military treatise written in China in 1628. Inside its 80 volumes are descriptions of many pyrotechnic devices. A rocket-powered arrow (firework) is described as having a range of 400 paces. An historian rings you up and asks you to estimate the maximum speed of these rockets. Applying the usual treatment of projectiles, what would your estimate be?

I don't really know where to start as there is no angle of launch (maybe 45 degrees?) or other data, apart from the distance being 400 paces (which is about 300m?)

What formula did your book give for this sort of exercise?

Note: This won't be the usual "projectile motion" formula, y = -0.5gt² + v₀t + h₀, because, as you've no doubt noticed, that formula always has the thing thrown or dropped; the formula won't apply to something that is launched and is still powered after launch, as are these rocket-propelled arrows. So we'll be needing whatever other formula they gave you.

By the way, "Mythbusters" actually tried this out. You can view the results online, such as here.