Two cars P and Q start from a point at the same time in a straight line and their positions are represented by and . At what time do the cars have the same velocity? [2016]

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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 [2016]

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

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

3 $\frac{a + f}{2 (1 + b)}$

4 $\frac{f - a}{2 (1 + b)}$

Ans.
(4)

Position of the car P at any time t, is

$x_{P} (t) = a t + b t^{2}; v_{P} (t) = \frac{d x_{P} (t)}{d t} = a + 2 b t$ ...(i)

Similarly, for car Q,

$x_{Q} (t) = f t - t^{2}; v_{Q} (t) = \frac{d x_{Q} (t)}{d t} = f - 2 t$ ...(ii)

$∵ v_{P} (t) = v_{Q} (t) (Given)$

$∴ a + 2 b t = f - 2 t or, 2 t (b + 1) = f - a$

$∴ t = \frac{f - a}{2 (1 + b)}$

Solution

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 [2016]

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

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

3 $\frac{a + f}{2 (1 + b)}$

4 $\frac{f - a}{2 (1 + b)}$

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Solution

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 [2016]

1 a-f1+b

2 a+f2(b-1)

3 a+f2(1+b)

4 f-a2(1+b)

Ans. (4) Position of the car P at any time t, is xP(t)=at+bt2; vP(t)=dxP(t)dt=a+2bt ...(i) Similarly, for car Q, xQ(t)=ft-t2; vQ(t)=dxQ(t)dt=f-2t ...(ii) ∵ vP(t)=vQ(t) (Given) ∴ a+2bt=f-2t or, 2t(b+1)=f-a ∴ t=f-a2(1+b)

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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 [2016]

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

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

3 $\frac{a + f}{2 (1 + b)}$

4 $\frac{f - a}{2 (1 + b)}$