Algebra / Calculus Problem?

Question

I am trying to prove that two components of motion equal another equation, but I am having trouble solving for the variable of one of them, the equation is

y = -16t^2 + Vsinxt

Where V is equal to a constant, it is supposed to be V-not, but I don't know how to type that. I am supposed to solve for t in the equation. Any help would be appreciated.

Answer 1

In my work, I let "V naught" (the initial velocity I presume?) be represented by V_o.

First, factor out -16: y = -16*[t^2 - (V_o sin x/16)t]

Add a squared term into the brackets to get an easily factorable expression, remembering to also add an equivalent term on the left side to maintain balance....

y - 16C^2 = -16*[t^2 - (V_o sin x/16)t + C^2]

The term in the brackets can now be easily written in the following form: (t - C)^2 = t^2 - 2Ct + C^2. Set the two coefficients for the "t" term equal and solve for C:

2C = V_o sin x/16 ------------> C = V_o sin x/32

Getting back to the original relationship, we now have:

(t - C)^2 = (y - 16C^2)/(-16) = (16C^2 - y)/16

Taking the square root of each side, we get:

(t - C) = +- (16C^2 - y)^0.5/4

t = C +- (16C^2 - y)^0.5/4

If you substitute the value of C into the expression on the right hand side and then simplify as much as possible, you will end up with this result:

t = (V_o sin x +- [(V_o sin x)^2 - 64y]^0.5)/32

Technically, there are two values of t represented by this equation (due to the "+-" sign) but if the variable represents time, obviously you can only have a positive solution for it. In other words, if the negative sign in front of the square root part ends up making the entire expression negative, discard that solution and use the remaining one.

I hope this helps.....

Answer 2

To solve this for t you must complete the square; first factor out the -16:

y = -16(t^2 + V sinx / 16 t)
then use half the coefficient of t, multiply it out and add the term that's subtracted

y = -16(t + V sinx/32)^2 + v^2 sin^2 x / 64

Now move the last part over by subtracting it, then divide both sides by -16, then sq rt both sides, then subtract the V sinx/32 from both sides

Good luck figuring all this out!