Alright, let's work through this step-by-step.
We start with the given formula for velocity:
[tex]\[
v = \frac{d_2 - d_1}{t_2 - t_1}
\][/tex]
To determine which option is equivalent, we should manipulate this original equation. First, let's remove the fraction by multiplying both sides by [tex]\((t_2 - t_1)\)[/tex]:
[tex]\[
v(t_2 - t_1) = d_2 - d_1
\][/tex]
Now, we have a simplified form of the equation. We'll compare our manipulated equation to each given option.
### Checking Option A:
[tex]\[
t_1 = v(d_2 - d_1) - t_2
\][/tex]
Substitute [tex]\(v = \frac{d_2 - d_1}{t_2 - t_1}\)[/tex]:
[tex]\[
t_1 = \left(\frac{d_2 - d_1}{t_2 - t_1}\right)(d_2 - d_1) - t_2
\][/tex]
Simplifying this does not give a meaningful equivalence. So, Option A is incorrect.
### Checking Option B:
[tex]\[
t_2 = v(d_2 - d_1) + t_1
\][/tex]
Substitute [tex]\(v = \frac{d_2 - d_1}{t_2 - t_1}\)[/tex]:
[tex]\[
t_2 = \left(\frac{d_2 - d_1}{t_2 - t_1}\right)(d_2 - d_1) + t_1
\][/tex]
This doesn't simplify cleanly either, so Option B is incorrect.
### Checking Option C:
[tex]\[
d_1 = v(t_2 - t_1) - d_2
\][/tex]
Substitute [tex]\(v = \frac{d_2 - d_1}{t_2 - t_1}\)[/tex]:
[tex]\[
d_1 = \frac{d_2 - d_1}{t_2 - t_1}(t_2 - t_1) - d_2
\][/tex]
Simplifying this yields:
[tex]\[
d_1 = (d_2 - d_1) - d_2
\][/tex]
[tex]\[
d_1 = d_2 - d_1 - d_2
\][/tex]
[tex]\[
d_1 = -d_1
\][/tex]
This is clearly incorrect. So, Option C is also incorrect.
### Checking Option D:
[tex]\[
d_2 = v(t_2 - t_1) + d_1
\][/tex]
Substitute [tex]\(v = \frac{d_2 - d_1}{t_2 - t_1}\)[/tex]:
[tex]\[
d_2 = \left(\frac{d_2 - d_1}{t_2 - t_1}\right)(t_2 - t_1) + d_1
\][/tex]
This simplifies to:
[tex]\[
d_2 = (d_2 - d_1) + d_1
\][/tex]
[tex]\[
d_2 = d_2 - d_1 + d_1
\][/tex]
[tex]\[
d_2 = d_2
\][/tex]
This is a true statement. Thus, Option D is correct.
So, the equivalent equation is:
[tex]\[
\boxed{d_2 = v(t_2 - t_1) + d_1}
\][/tex]
Therefore, the correct answer is [tex]\( \text{Option D} \)[/tex].
