I am trying to derive an equation. We are given the solutions to two equations,

Question

I am trying to derive an equation. We are given the solutions to two equations, which should look like this: a =2 x x2/t2 - x1/t1 divided by t2-t1 and vo = (t2/t1)x1 -(t1/t2)x2 divided by t2- t1 from the kinematic equation "x= vo + 1/2at^2 (I am having trouble deriving the vo = equation) The professor provided a hint by stating this "Since you are given two sets of positions (x1 and x2) and times (t1 and t2) these values are considered "knowns". To solve for two unknowns you should realize that you will need at least two different equations. Which of three kinematics equations should I use: Select the kinematic equation ( x(t)=x0 + v0t + 1/2at2 ) because it already contains both x, and t and the unknown variables (v0 and a) that you are trying to solve. So you must solve two simultaneous equations for a and v0. x1=x0 + v0t1 + 1/2at12 and x2=x0 + v0t2 + 1/2at22 Think back to algebra class and possibly three different approaches (all valid) to solving a system of simultaneous equations. ie., solving each equation for the variable you're trying to eliminate, and then setting the two results equal to each other, or solving one equation for one of the variables, and then substituting it into the second equation., or adding or substracting one equation from the other in such a way that you eliminate one of the variables (may need to factor out something or multiply by something in order to have this work). Caveats - remember to use common denominations when adding and subtracting. need to do the same to both sides of an equation. remembering that you may have both positive and negative roots. Although in theory - any of the three methods should work equally well, in practice- one of the methods may be easier than the others for a particular person. If you get stuck using one method, try a different one, before you get frustrated!" I understand how to get the first equation for acceleration now, but I am confused about how to derive the equation for vo. Please help.

Explanation / Answer

v= vo+at x1=vot1+1/2 at1^2 x2=vot2 +1/2 at2^2 x1t2=vot1t2+1/2at1^2 t2 x2t1=vot2t1+1/2 at2^2 t1 subtracting the above equations gives x1t2-x2t1=1/2a(t1t2(t1+t2)) hence a =2(x1t2-x2t1)/(t1t2(t1+t2)) substituting in the above equation gives vo

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