Maximum Distance Traveled with Constant Propulsion

[ciao_potter]

Lets say that we have a rocket positioned at a certain angle

with respect to the horizontal. The rocket is fired, with a constant propulsion force of

. Assume that the rocket has a constant mass of

kg (the gas used to propel the rocket is negligibly small). What is the optimal angle

with which the rocket must be fired to have the farthest distance covered when it lands on the ground?

You can view my attempt https://latex.artofproblemsolving.com/miscpdf/eyecbxwg.pdf?t=1500934067562.

The differentiating process to find the max value of x is very messy and requires a calculator.

I was wondering if there are any better approaches to this problem.

 

[.Scott]

Your rocket will never leave the ground - even if you point it straight vertically into the air. So it makes no difference.

Here's what could change it:
1) put wings on it.
2) Increase the propulsion so that the acceleration is more than 1G. Then point it straight up and it will eventually achieve escape velocity.

 

[ciao_potter]

Can you explain to me why it will not leave the ground?

 

[gneill]

Can you explain to me why it will not leave the ground?

You haven't accounted for the weight of the rocket (gravitational force). Draw the FBD.

 

[Ray Vickson]

Lets say that we have a rocket positioned at a certain angle View attachment 207744 with respect to the horizontal. The rocket is fired, with a constant propulsion force of View attachment 207745. Assume that the rocket has a constant mass of View attachment 207746 kg (the gas used to propel the rocket is negligibly small). What is the optimal angle View attachment 207747 with which the rocket must be fired to have the farthest distance covered when it lands on the ground?

You can view my attempt https://latex.artofproblemsolving.com/miscpdf/eyecbxwg.pdf?t=1500934067562.

The differentiating process to find the max value of x is very messy and requires a calculator.

I was wondering if there are any better approaches to this problem.

I think the problem is ill-conceived. Here is why:

How does your rocket get launched? Is it on a catapult that hurls it up at initial vertical speed $v_{0y}$, or is it launched from rest like most rockets? If the latter, you have initial $x$ and $y$ velocities equal to zero, but horizontal acceleration $a_x = a \cos \alpha$ and vertical acceleration $a_y = a \sin \alpha - g$, where $g$ is the acceleration of gravity ($g \doteq 9.8$ ($m/\text{sec}^2$). Anyway, if the angle $\alpha$ remains constant your rocket will travel in a straight line until its fuel runs out; then it will go into free-fall, but from a nonzero $(x,y)$ location with $y > 0$ and with some nonzero initial velocities $v_x$ and $v_y$. Its total $x$-range is the distance from the starting point $(0,0)$ and the final point where $y=0$ again at the end of the free-fall phase.

It is possible to obtain an expression for the $x$-range in terms of $\alpha$ and $T$ (the total fuel burn-time), but maximizing it "analytically" is difficult. However, given a specific burn-time, the problem can be solved numerically using an optimization package (such as the EXCEL solver, for example).