# Rocket Max Height/Time Problem

## Homework Statement

A 0.2-lb model rocket is launched vertically from rest at time t=0 with a constant thrust of 2 lb for one second and no thrust for t > 1 sec. Neglecting air resistance and the decrease in mass of the rocket, determine (a) the maximum height h reached by the rocket, (b) the time required to reach this maximum height.

Weight = 0.2 lb, Thrust force = 2 lb/sec

(1) FT-mg = ma
W = mg

## The Attempt at a Solution

I solved for the mass of the rocket by using m = W/g, where W = 0.2 lb and g = 32.2 ft/sec^2. Then I solved for 'a' in Equation (1) to get 289.8 ft/sec^2. To find the maximum height, I used y = y0 + v0t + 1/2*(at)^2. Since v0 = 0, I got y-y0 = 144.9 ft. (I think the correct answer is supposed to be 1449 ft - looks like I am off by a factor of 10).

Also, for the second part, how do you find the time?

I think I'm missing something here...

Thank you

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cristo
Staff Emeritus
What first jumps to mind is that the pound is a unit of mass, isn't it? Could you show your work, i.e. how exactly you got 144.9ft, and we can check if you're correct.

Yes, do I need to convert the thrust to Newtons? 1 lb = 4.448 N

m = W/g = 0.2 lb/32.2 ft/sec^2

a = (Ft-mg)/m = [(2 lb - 0.006211 (lb*sec^2/ft)(32.2 ft/sec^2)]/[0.006211 lb*s^2/ft]

v0 = 0

y-y0=v0t+1/2(at)^2

y-y0 = 1/2(289.8 ft/sec^2)(1 sec)^2

y-y0 = 144.9 ft

D H
Staff Emeritus
You do not need to convert to metric. You can do everything in English units. $F=ma$ still applies in Enlish units with force in pounds, mass in slugs, and acceleration in feet/second/second.

Your mistake is assuming the vehicle reaches maximum height when the rocket stops thrusting.

cristo
Staff Emeritus
You do not need to convert to metric. You can do everything in English units. $F=ma$ still applies in Enlish units with force in pounds, mass in slugs, and acceleration in feet/second/second.

What's a slug? This might just be me, but isn't a pound a unit of mass? For example, when I go to the greengrocers, produce is sold in pounds or kilograms. When I learned imperial units in school, we were taught that pounds and ounces are units of mass?!

A pound is a measure of gravitational force and not mass. A lot of people don't know the difference between weight and mass, so you were probably just taught wrong in school. I really wish that the bill (during Carter years?) making everything metric would have passed. For some reason mathematicians like English units, but I forget why (something about base 12).

ranger
Gold Member
I believe that 1 pound force = 4.45 newtons. Slug is the unit of mass for the English system. It is a mass that accelerates by 1 ft/s² when a force of one pound-force (lbf) is exerted on it
http://en.wikipedia.org/wiki/Slug_(mass [Broken])

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cristo
Staff Emeritus
Well, I'm from England, and was taught in Physics and in Maths at school that a pound is a unit of mass. A pound is equal to 12 ounces, and an ounce is ~28grams. See here:http://www.engineeringtoolbox.com/british-imperial-system-d_993.html or here: http://en.wikipedia.org/wiki/Imperial_unit

There seems to be some sort of ambuguity here, since the wiki article appears to use the terms mass and weight interchangably. However, I stand by that fact that, given I was educated by physicists who used imperial units in everyday life, that a pound is a unit of mass!!

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D H
Staff Emeritus
The pound is both a unit of mass (better specified as pound-mass, abbreviated as lbm) and a unit of force (better specified as pound-force, abrbeviated as lbf). On an idealized surface of the Earth, an object with a mass of 1 lbm has a weight of 1 lbf. Weight is a force.

The original poster used pounds as a force when talking about the thrust of the rocket. Whether the rocket had a mass of 0.2 lbm or weight of 0.2 lbf was not specified.

Other than the confusing nomenclature (lbm vs. lbf), another big problem with expressing mass in lbm and force in lbf is that F=ma does not apply. A constant of proportionality is needed.

Having F=ma (as opposed to F=kma) is a very nice feature. There are two ways to do this and still use some sort of English units: use pound-mass for mass, and something other than pound-force for force, or use pound-force for force and something other than pound-mass for mass. These alternate units are the poundal, the force needed to make a mass of one pound-mass accelerate by one foot/second2, and the slug, the mass that makes an object subject to a force of one pound-force accelerate by one foot/second2.

cristo
Staff Emeritus
Well, that seems a pretty stupid system where two names mean different things! Anyway, sorry for my hijacking; I've just never used pound to be a unit of force before! Whenever we used the units (which was rarely to never) we just used lb ft/s2 as the force!

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D H
Staff Emeritus
Turbine engine manufacturers (even European ones) often express thrust in pounds, with kilonewtons as an afterthought.

D H
Staff Emeritus
Back on topic,

Lilshai, you had the correct height for the height of the rocket at the time thrust terminates. What happens to the rocket after the engine stops?

"Your mistake is assuming the vehicle reaches maximum height when the rocket stops thrusting."

How come? After the engine stops, the rocket would fall back down rite...so wouldn't the max height be when v = 0, or when the thrust terminates?

D H
Staff Emeritus