Question 79: Motion in 1D

Question

In a car race on straight road, car A takes a time t less than car B at the finish and passes finishing point with a speed '$$\upsilon $$' more than that of car B. Both the cars start from rest and travel with constant acceleration a
1
and a
2
respectively. Then '$$\upsilon $$' is equal  to :

Accepted Answer

For both car initial speed ($$\mu $$) = 0
Let the acceleration of car A and car B is $$a$$
1
and  $$a$$
2
respectively.
Also let the time taken to reach the finishing point for car A is t
1
and for car B is t
2
.
Let at finishing point speed of car A is $$v$$
1
and speed of car B is $$v$$
2
According to the question,
t
2
$$-$$ t
1
= t
and   $$v$$
1
$$-$$ $$v$$
2
= $$v$$
$$ \Rightarrow $$  $$a$$
1
t
1
$$-$$ $$a$$
2
t
2
= $$v$$
$$ \Rightarrow $$  $$a$$
1
t
1
$$-$$ $$a$$
2
(t + t
1
) = $$v$$ . . . . . . .(1)
As, Total distance covered by both car is equal.
So,   x
A
= x
B
$$ \Rightarrow $$  $${1 \over 2}{a_1}t_1^2 = {1 \over 2}{a_2}t_2^2$$
$$ \Rightarrow $$  $$a$$
1
t$$_1^2$$ = $$a$$
2
(t + t
1
)
2
$$ \Rightarrow $$  $$\sqrt {{a_1}} .{t_1}$$ = $$\sqrt {{a_2}} $$ . (t + t
1
)
$$ \Rightarrow $$  $$\sqrt {{a_1}} .{t_1} - \sqrt {{a_2}} .{t_1} = \sqrt {{a_2}} .t$$
$$ \Rightarrow $$  t
1
= $${{\sqrt {{a_2}} .t} \over {\sqrt {{a_1}}  - \sqrt {{a_2}} }}\,\,\,\,\,\,\,.....(2)$$
Now put the value of t
1
in equation (2),
($$a$$
1
$$-$$ $$a$$
2
) t
1
$$-$$ $$a$$
2
t = $$v$$
$$ \Rightarrow $$  (a
1
$$-$$ a
2
) . $${{\sqrt {{a_2}} .t} \over {\sqrt {{a_1}}  - \sqrt {{a_2}} }} - {a_2}t = v$$
$$ \Rightarrow $$  $$\left( {\sqrt {{a_1}}  + \sqrt {{a_2}} } \right)\sqrt {{a_2}} .t - {a_2}t = v$$
$$ \Rightarrow $$  $$\sqrt {{a_1}{a_2}} .t + {a_2}.t - {a_2}t = v$$
$$ \Rightarrow $$  $$v$$ = $$\sqrt {{a_1}{a_2}} .t$$

Answer

$${{2{a_1}{a_2}} \over {{a_1} + {a_2}}}t$$

Answer

$$\sqrt {2{a_1}{a_2}} t$$

Answer

$$\sqrt {{a_1}{a_2}} t$$

Answer

$${{{a_1} + {a_2}} \over 2}t$$

Motion in 1D

In a car race on straight road, car A takes a time t less than car B at the finish and passes finishing point with a speed '$\upsilon ' more than that of car B. Both the cars start from rest and travel with constant acceleration a 1 and a 2 respectively. Then '\upsilon $' is equal to :

Choose the correct answer: