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In Heather's physics class, she had to solve the equation [tex][tex]$7=\frac{1}{2} a t^2+u t$[/tex][/tex] for [tex][tex]$u$[/tex][/tex]. Which equation correctly solves for [tex][tex]$u$[/tex][/tex]?

A. [tex]u=\frac{14-a t^2}{t}[/tex]
B. [tex]u=7-\frac{1}{2} a t^2-t[/tex]
C. [tex]u=\frac{7-\frac{1}{2} a t^2}{t}[/tex]
D. [tex]u=14-a t^2-2 t[/tex]



Answer :

GinnyAnswer
Sure, let's solve the given equation step by step to determine which equation correctly solves for [tex]\( u \)[/tex].

We start with the equation:
[tex]\[ 7 = \frac{1}{2} a t^2 + u t \][/tex]

Our goal is to solve for [tex]\( u \)[/tex].

1. Step 1: Isolate the term involving [tex]\( u \)[/tex] on one side

Subtract [tex]\(\frac{1}{2} a t^2\)[/tex] from both sides to move it away from [tex]\( u t \)[/tex]:
[tex]\[ 7 - \frac{1}{2} a t^2 = u t \][/tex]

2. Step 2: Solve for [tex]\( u \)[/tex]

To isolate [tex]\( u \)[/tex], divide both sides of the equation by [tex]\( t \)[/tex]:
[tex]\[ u = \frac{7 - \frac{1}{2} a t^2}{t} \][/tex]

So, the equation that correctly solves for [tex]\( u \)[/tex] is:
[tex]\[ u = \frac{7 - \frac{1}{2} a t^2}{t} \][/tex]

Now, let's compare this solution to the given options:

1. [tex]\( u = \frac{14 - a t^2}{t} \)[/tex]

2. [tex]\( u = 7 - \frac{1}{2} a t^2 - t \)[/tex]

3. [tex]\( u = \frac{7 - \frac{1}{2} a t^2}{t} \)[/tex]

4. [tex]\( u = 14 - a t^2 - 2 t \)[/tex]

Clearly, option (3) matches our derived solution. Therefore, the correct equation that solves for [tex]\( u \)[/tex] is:
[tex]\[ u = \frac{7 - \frac{1}{2} a t^2}{t} \][/tex]

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