To solve for acceleration, we need to identify an equation that isolates the acceleration (denoted as [tex]\( a \)[/tex]) in terms of other variables. Let's examine each of the given equations:
1. [tex]\( t = \frac{\Delta v}{a} \)[/tex]:
- Rearranging the equation to solve for [tex]\( a \)[/tex]:
[tex]\[
a = \frac{\Delta v}{t}
\][/tex]
This equation correctly represents formula for acceleration where [tex]\( \Delta v \)[/tex] is the change in velocity and [tex]\( t \)[/tex] is the time over which the change occurs.
2. [tex]\( v = a t - n \)[/tex]:
- This equation represents a relationship between velocity [tex]\( v \)[/tex], acceleration [tex]\( a \)[/tex], time [tex]\( t \)[/tex], and some constant [tex]\( n \)[/tex]. It does not directly isolate [tex]\( a \)[/tex] in a straightforward manner for solving for acceleration.
3. [tex]\( a = \frac{d}{t} \)[/tex]:
- This equation suggests that acceleration [tex]\( a \)[/tex] is equal to distance [tex]\( d \)[/tex] divided by time [tex]\( t \)[/tex]. However, this is incorrect as acceleration is not given by distance divided by time; rather, acceleration involves the change in velocity over time.
4. [tex]\( \Delta v = \frac{a}{t} \)[/tex]:
- Rearranging the equation to solve for [tex]\( a \)[/tex]:
[tex]\[
a = \Delta v \cdot t
\][/tex]
This form does not correctly describe the acceleration. Instead, it should be that [tex]\( \Delta v = a \cdot t \)[/tex], aligning more accurately with [tex]\( a = \frac{\Delta v}{t} \)[/tex].
Given the analysis of these equations, the correct equation that can be used to solve for acceleration is:
[tex]\[
a = \frac{\Delta v}{t}
\][/tex]
This correctly aligns with the standard definition of acceleration, which is the change in velocity ([tex]\( \Delta v \)[/tex]) divided by the time interval ([tex]\( t \)[/tex]) over which the change occurs. Hence, the choice is:
[tex]\[ t = \frac{\Delta v}{a} \][/tex]
(rearranged as [tex]\( a = \frac{\Delta v}{t} \)[/tex]).
