Question

On a two-lane road, car A is travelling with a speed of 36 km/hr. Two cars B and C approach car A in opposite directions with a speed of 54 km/hr each. At a certain instant, when the distance AB is equal to AC, both being 1 km, B decides to overtake A before C does. What minimum acceleration of car B is required to avoid an accident ?​

Answer 1

Given :

Velocity of car A, V$\sf _A$ = 36km/hr = 10m/s

Velocity of car B, V$\sf _B$ = 54km/hr = 15m/s

Velocity of car C, V$\sf _C$ = 54km/hr = 15m/s

To find :

The minimum acceleration of car B is required to avoid an accident.

Solution :

First we have to find relative velocity of car B with respect to car A and car C with respect to car A.

» Velocity of one body with respect to other body is called relative velocity.

◈ Relative velocity of car B with respect to car A,

$\dashrightarrow \sf V_{BA} = V_B - V_A$

$\dashrightarrow \sf V_{BA} =15 - 10$

$\dashrightarrow \sf V_{BA} = 5 {ms}^{ - 1}$

◈ Relative velocity of car C with respect to car A,

$\dashrightarrow \sf V_{CA} = V_C -( - V_A)$

$\dashrightarrow \sf V_{CA} = 15 + 10$

$\dashrightarrow \sf V_{CA}= 25 {ms}^{ - 1}$

At the certain instance, both cars B and C are the same distance from car A that is,

s = 1km/hr = 1000 m

Time taken by the car C to cover 1000m = 1000/25 = 40sec.

thus, to avoid an accident, car B must cover the same distance in a maximum of 40sec.

Now, using second equation of kenimatics that is,

$\boxed{\bf s = ut + \dfrac{1}{2}at^2}$

where,

• s denotes distance covered
• u denotes initial velocity
• t denotes time taken
• a denotes acceleration

$\leadsto\sf s = ut + \dfrac{1}{2} at^2$

$\leadsto\sf 1000 = (5)(40) + \dfrac{1}{2} \times a \times (40)^2$

$\leadsto\sf 1000 = 200 + \dfrac{1}{2} a \times 1600$

$\leadsto\sf 1000 - 200 = \dfrac{1}{2} \times a \times 1600$

$\leadsto\sf 800 = \dfrac{1}{2} \times a \times 1600$

$\leadsto\sf 1600 = 1600a$

$\leadsto\sf a = \dfrac{1600}{1600}$

$\leadsto\underline{\boxed{\sf a = 1 {ms}^{ - 2}}}$

thus, the minimum acceleration of car B is required to avoid an accident is 1 m/s²

Answer 2

Please refer to the above picture