dimensional analysis?

Question

One might expect that the product of the mass m times the velocity V of some object would depend on both the external force F applied to the object and the time interval t during which this force is applied, i.e.
mV = F^a·T^b
where a and b are constants to be determined. Using dimensional analysis, show that the correct values of a and b are
1. a=1, b=1/2
2. a=1/2, b=1

I don't get what they're trying to make me find out

a=2, b=-1
a=1, b=1
a=1, b=2

these were the rest of the options..i just wanted to know what the question meant..thanks for the help though

Answer 1

Pretty much what they are trying to make you do is show that you can intuitively derive Newton's first law of motion in its original mathematical form.

You are correct about assuming dimensional analysis - look at all of the units involved, and see how you could get them to be the same on each side of the equals sign by using the powers suggested by the worksheet.

m = kg
v = m/s
F = kgm/s^2
t = s

Personally, I don't see how either of those answers work as the equation is mass times delta v = force times delta t which works out in terms of units. Ask your teacher about that one...it doesn't seem right to me as both a and b should equal 1.

Hope that means something...ask if it doesn't.

Answer 2

Dimensional analysis is basically the process of following the units of your quantities as you go through your equation. It is most useful in conversion operations - e.g. if you want to turn 5 seconds into hours you do ( 5s/1 ) * ( 1min/60s ) * ( 1hr/60min ). Essentially, you want all your units in these divisions and multiplications to cancel out, except for the one unit you want to end up with (in this case, hours). In your second question, you need to look at the units. I'll just assume some of the more common ones as an example. m = meters, s = seconds s = k * a^m * t^n m = k * ( (m/s^2)^1 ) * ( s^2 ) Combining the stuff on the right side: m = (k * m * s^2) / (s^2) You can see that the seconds cancel out in the division, so you are left with: m = k * m Solving for k, you end up with 1. Since there are no units here, you can see that k must be a dimensionless constant! Hope that made sense!

Answer 3

first you need to work out all the dimensions for mass, velocity, force and time

the idea of dimensional analysis is to work out formulas, when you only know roughly what they depend on, ie you know here that the momentum (mass * velocity) is related to force and time, but not directly. dimensional analysis should enable you to work out how these things are related by helping you to calculate the powers a and b

first turn your force into mass * acceleration, then on the left hand side you've got

mass * velocity which is kg * metres / seconds

and on the right hand side you've got

mass * acceleration * time which is kg * metres / seconds / seconds * seconds

both kg and metres have the same power on each side, as does seconds, because on the right hand side two of the seconds cancel. so the correct values of a and b are 1 and 1 because

m v = F t

hmmm, either i've gone wrong or the questions wrong, i think its the question because, mv is momentum and Ft is impulse and i know they've got the same units

hope that helps

Answer 4

What the question implies is that you can find a and b through dimensional analysis. DA deals with the units (metrics) that attend the variables in equations.

For example, your mV => kg m/sec in kms SI units. FT = maT => kg (m/sec^2) sec = kg m/sec as well. Thus, we can conclude the variables on either side of the mV = FT equation are in fact equivalent from a physics point of view. That is, both sides represent momentum, which has units kg m/sec.

Since the units are equivalent as is on both sides, we can conclude that a = 1 and b = 1 for your question.

Units analysis is a great way to check your equations. If they are incorrect, that will show up by an imbalance of the units on the left and right-hand sides of the equals sign.

Answer 5

It's a units question.
Get the force and time units
in the same form as the mass and
velocity units. Then you will see that to
make the units on each side identical you
will need to raise force and time to certain powers.
Make sense?