Question

a student is manipulating the equation p1v1/t1 = p2v2/t2 to solve for t2. they want to avoid common algebraic errors. which step - by - step approach minimizes the chance of mistakes? invert both sides first, then cross multiply multiply both sides by t1t2 first, then rearrange multiply both sides by t2 only, then divide by the left side cross multiply immediately: p1v1t2 = p2v2t1, then divide by p2v2 a physics student needs to solve for acceleration (a) from the equation f = ma + bv², where f is force, m is mass, v is velocity, and b is a drag coefficient. which rearrangement is correct? a=(f - bv²)/m a = f/(m + bv²) a = f - bv²/m a = f - ma - bv²

Explanation:

Step1: Solve the first - equation

We start with the equation $\frac{P_1V_1}{T_1}=\frac{P_2V_2}{T_2}$. Cross - multiplying immediately gives us $P_1V_1T_2 = P_2V_2T_1$. Then dividing both sides by $P_2V_2$ isolates $T_2$ as $T_2=\frac{P_2V_2T_1}{P_1V_1}$. This is a straightforward way to solve for $T_2$ and minimizes the chance of making algebraic errors.

Step2: Solve the second - equation

We start with $F = ma+ bv^2$. First, we want to isolate the term with $a$. We subtract $bv^2$ from both sides of the equation: $F - bv^2=ma$. Then, to solve for $a$, we divide both sides by $m$. So, $a=\frac{F - bv^2}{m}$.

Answer:

1. Cross multiply immediately: $P_1V_1T_2 = P_2V_2T_1$, then divide by $P_2V_2$
2. $a=(F - bv^2)/m$