Answer: first and third.
Explanation:
An equation is dimensionally correct if the units are the same in both sides of the equation.
first, let's define the units used:
{m} = kg
{v} = m/s
{F} = kg*m/s^2
{x} = m
{t} = s
{a} = m/s^2
Now, let's analyze each option:
1) m*v/t = F
in the left side the units are:
{m}*{v}/{t} = kg*(m/s)*(1/s) = kg*m/s^2
And as is written above, these are the units of F, so this is correct.
2) x*v^2 = F*(x^3/x^2)
This is more trivial, in the right side we can see an F, that has mass units (kg) and in the left side we have x and v, and we know that none of these have mass units, so this expression is not correct.
3) xt= vt^2+at^3
the units in the right side are:
{x}*{t] = m*s
in the right side are:
{v}*{t}^2 + {a}*{t}^2 = (m/s)*s^2 + (m/s^2)*s^3 = m*s + m*s
So in both sides of the equation we have the same units, then this equation is dimensionally correct.
