Two cars P and Q start from a point at the same time in a straight line and their positions are represented by x P ( t )= at + bt 2 and x Q ( t )=f t - t 2. At what time do the cars have the same velocity ?

Question

Two cars P and Q start from a point at the same time in a straight line and their positions are represented by x P ( t )= at + bt 2 and x Q ( t )=f t - t 2. At what time do the cars have the same velocity ? (A) (a - f/1 + b) (B) (a + f/2( b - 1)) (C) (a + f/2( 1 + b)) (D) (f - a /2( 1 + b))

Tardigrade Admin · Accepted Answer

vp = (dxp/dt) = a +2bt vQ = (dxQ/dt) = f - 2t vP = vQ ⇒ a +2bt =f-2t t =( f-a/2(b+1))

Q. Two cars $P$ and $Q$ start from a point at the same time in a straight line and their positions are represented by $x_{P} (t) = a t + b t^{2}$ and $x_{Q} (t) = f t - t^{2}$ . At what time do the cars have the same velocity ?

$\frac{a - f}{1 + b}$

$\frac{a + f}{2 ( b - 1 )}$

$\frac{a + f}{2 ( 1 + b )}$

$\frac{f - a}{2 ( 1 + b )}$

$v_{p} = \frac{d x _{p}}{d t} = a + 2 b t$
$v_{Q} = \frac{d x _{Q}}{d t} = f - 2 t$
$v_{P} = v_{Q} \Rightarrow a + 2 b t = f - 2 t$
$t = \frac{f - a}{2 ( b + 1 )}$

Q. Two cars P and Q start from a point at the same time in a straight line and their positions are represented by xP​(t)=at+bt2 and xQ​(t)=ft−t2. At what time do the cars have the same velocity ?

1+ba−f​

2(b−1)a+f​

2(1+b)a+f​

2(1+b)f−a​