Question

two cars are moving from opposite direction with velocities v1 and v2. when the distance between them was S, brakes were applied by both the cars as a result of which retardations produced in them become a1 and a2 respectively. Show that there will be no collision between the cars if 2a1a2S is greater than or equal to v1^2a2 + v2^2a1.​

Answer 1

Answer:

hey

Explanation:

Formula used:

v2=u2+2as

Complete answer:

In the question it is said that two cars, car 1 and car 2 are moving in the same direction on a straight road with different velocities.

The initial velocity of the car 1 is ‘v1

’ and of car 2 is ‘v2

’.

It is given that v1>v2

The distance between the two cars is given as ‘d’.

The retardation of car 1 when the driver applies break is ‘a1’ and the acceleration of car 2 is ‘a2’.

From the third equation of motion, the relative motion between the two cars can be written as,

v2rel=u2rel+2arelsrel

, were ‘v’ is final velocity, ‘u’ is initial velocity, ‘a’ is acceleration and ‘s’ is the separation.

Let ‘v′1

’ be the final velocity of the car 1 and ‘v′2

’ be the final velocity of car 2.

Then the relative final velocity can be written as,

vrel=v′1−v′2

And the relative initial velocity can be written as,

urel=v1−v2

Relative acceleration will be,

arel=−a1−a2

⇒arel=−(a1+a2)

And the relative separation,

srel=−d

By substituting this in these values in the relative equation, we get

⇒(v′1−v′2)2=(v1−v2)2+2(−(a1+a2))(−d)

⇒(v′1−v′2)2=(v1−v2)2+2(a1+a2)(d)

In the question we are asked to find the minimum initial separation to avoid collision.

We know that to avoid collision the final velocity of car 1 should be less than the final velocity of car 2, i.e. (v′1−v′2)

By applying this condition in the equation, we get the minimum distance as,

⇒d=(v1−v2)2−(v′1−v′2)22(a1+a2)

This is the minimum distance to avoid collision.

Note:

Relative motion of a body is the motion of that body with respect to some other, adjacent frame of reference. Every motion will always be relative to some frame of reference.

If there are two objects A and B, the relative motion of the object A with respect to B is said as the rate of change of position of A with respect to the object B.

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