Question

which equation can be used to solve for acceleration?
$t=\frac{delta v}{a}$
$vf = at - vi$
$a=\frac{d}{t}$
$delta v=\frac{a}{t}$

Explanation:

Step1: Recall acceleration formula

The basic formula for acceleration \( a \) is \( a=\frac{\Delta v}{t} \), where \( \Delta v \) is change in velocity and \( t \) is time. Let's analyze each option:

Step2: Analyze Option 1 (\( t = \frac{\Delta v}{a} \))

Starting with \( t=\frac{\Delta v}{a} \), we can rearrange it to solve for \( a \). Multiply both sides by \( a \): \( t\times a=\Delta v \), then divide both sides by \( t \): \( a = \frac{\Delta v}{t} \). Wait, no, wait. Wait, the first option is \( t=\frac{\Delta v}{a} \). Let's solve for \( a \): Cross - multiply, \( a\times t=\Delta v \), so \( a=\frac{\Delta v}{t} \)? Wait, no, \( t=\frac{\Delta v}{a}\Rightarrow a=\frac{\Delta v}{t} \)? Wait, no, let's do it correctly. If \( t=\frac{\Delta v}{a} \), then multiply both sides by \( a \): \( a\times t=\Delta v \), then divide both sides by \( t \): \( a = \frac{\Delta v}{t} \). Wait, but let's check other options.

Step3: Analyze Option 2 (\( v_f=at - v_i \))

Rearrange \( v_f = at - v_i \): Add \( v_i \) to both sides: \( v_f + v_i=at \), then divide by \( t \): \( a=\frac{v_f + v_i}{t} \), but \( \Delta v=v_f - v_i \), so this is not the standard acceleration formula.

Step4: Analyze Option 3 (\( a=\frac{d}{t} \))

\( \frac{d}{t} \) is the formula for speed (distance over time), not acceleration. So this is incorrect.

Step5: Analyze Option 4 (\( \Delta v=\frac{a}{t} \))

Rearrange \( \Delta v=\frac{a}{t} \): Multiply both sides by \( t \): \( \Delta v\times t=a \), which would mean \( a = \Delta v\times t \), which is not the correct acceleration formula.

Wait, going back to Option 1: \( t=\frac{\Delta v}{a} \). Let's solve for \( a \). We know that acceleration \( a=\frac{\Delta v}{t} \) (from the definition \( a=\frac{v_f - v_i}{t}=\frac{\Delta v}{t} \)). If we have \( t=\frac{\Delta v}{a} \), we can re - arrange it to \( a=\frac{\Delta v}{t} \) (by cross - multiplying: \( a\times t=\Delta v\Rightarrow a = \frac{\Delta v}{t} \)). Wait, maybe I made a mistake earlier. Let's re - express the options:

The correct formula for acceleration is \( a=\frac{\Delta v}{t} \), which can be derived from \( t=\frac{\Delta v}{a} \) (by solving for \( a \)). Let's check each option again:

• Option 1: \( t=\frac{\Delta v}{a} \). Solving for \( a \): \( a=\frac{\Delta v}{t} \), which is the correct acceleration formula.
• Option 2: \( v_f=at - v_i \) gives \( a=\frac{v_f + v_i}{t} \), which is wrong.
• Option 3: \( a=\frac{d}{t} \) is speed, wrong.
• Option 4: \( \Delta v=\frac{a}{t} \) gives \( a=\Delta v\times t \), wrong.

So the equation \( t = \frac{\Delta v}{a} \) can be used to solve for acceleration (by rearranging it to \( a=\frac{\Delta v}{t} \)).

Answer:

The first option: \( \boldsymbol{t=\frac{\Delta v}{a}} \) (when rearranged, it gives the acceleration formula \( a = \frac{\Delta v}{t} \))