Expert: Daniel Mazur - 6/16/2009

Question
ACCELERATIONS for SAME FORCE on DIFFERENT MASSES

A force F of 1.98 N acting on a mass m1 results in an acceleration of 0.99 m/s2.
The same force F acting on mass m2 results in an acceleration of 3.00 m/s2.
What is the acceleration a3 for F pushing the combination of m1 and m2?

Pleas, show me how to solve out this problem, not only the answer.

Answer
Hello Bo Ram,

this is not a complicated problem, I'll be glad to give you a push in the right direction. It all relies in the Newton's law in the form F=m*a. According to this template we can write the same law for all three cases considered in the problem.
(1)   F=m1*a1      ,
(2)   F=m2*a2      and
(3)   F=(m1+m2)*a3 .

It is stated in the problem that F=1.98N is in each case the same and you are also given numerical values of a1=0.99m/s^2 and a2=3.00m/s^2. Using these values you can calculate the mass values
(1*)   m1=F/a1 = 2.0kg
(2*)   m2=F/a2 = 0.66kg
and these in turn allow you to solve the last equation
(3*)   a3=F/(m1+m2) = 1.98N/2.66kg = 0.74m/s^2 .

Another approach is to solve the problem symbolically till the end. Take equations (1*) and (2*) and substitute them symbolically (!) to eq. (3). The result will be
(4)   F=(m1+m2)*a3 = (F/a1+F/a2)*a3 = F*a3*(1/a1+1/a2) = F*a3*(a1+a2)/(a1*a2)
From here the force F drops and we can write
(5)   1=a3*(a1+a2)/(a1*a2), which leads to
(6)   a3=a1*a2/(a1+a2) = (2.97m^2/s^4)/(3.99m/s^2)= 0.74m/s^2 .

In order to get this result, all you need is to know how to solve systems of 3 linear equations (with all the mathematical operations involved). If you are not confident enough in calculus, please practice - perfect calculus skill is essential for good physics problem solving.

The nice aspect about the symbolic solution (equation (6)) is that it shows you that you don't need to know the value of force F to solve the problem! This is an important advantage in more complicated problems, so a full symbolic solution is always preferable before we plug in numbers.

Cheers!
Daniel