a car a is travelled with a speed of 72km/h on a straight horizontal road.it is followed by another car b which is moving with a velocity of 36km/h.when the distance between them is 25km,the car a is given deceleration of 2m/s^2.after

To analyze the situation with the two cars, we need to break down the problem step by step. Car A is initially traveling at a speed of 72 km/h, while Car B is moving at 36 km/h. When the distance between them is 25 km, Car A begins to decelerate at a rate of 2 m/s². Our goal is to determine how long it will take for Car B to catch up to Car A after this deceleration begins.

Converting Units

First, let's convert the speeds from kilometers per hour to meters per second, as this will make our calculations easier. The conversion factor is:

• 1 km/h = 1/3.6 m/s

So, for Car A:

• 72 km/h = 72 / 3.6 = 20 m/s

And for Car B:

• 36 km/h = 36 / 3.6 = 10 m/s

Initial Conditions

At the moment Car A starts to decelerate, its speed is 20 m/s, and Car B is traveling at 10 m/s. The distance between them is 25 km, which is equivalent to 25,000 meters.

Deceleration of Car A

Car A is decelerating at 2 m/s². We can calculate the time it takes for Car A to come to a stop using the formula:

v = u + at

Where:

• v = final velocity (0 m/s when it stops)
• u = initial velocity (20 m/s)
• a = acceleration (which is negative in this case, so -2 m/s²)

Rearranging the formula gives us:

t = (v - u) / a

Substituting in the values:

t = (0 - 20) / -2 = 10 seconds

Distance Covered by Car A

During this time, we can calculate how far Car A travels before it stops. Using the formula for distance:

s = ut + 0.5at²

Substituting the values:

• s = 20 * 10 + 0.5 * (-2) * (10)²
• s = 200 - 100 = 100 meters

Distance Between the Cars After 10 Seconds

Initially, the distance between the cars was 25,000 meters. After 10 seconds, Car A has traveled 100 meters, so the new distance between them is:

Distance = 25,000 - 100 = 24,900 meters

Distance Covered by Car B in 10 Seconds

Now, let's calculate how far Car B travels in the same 10 seconds:

Distance = speed × time = 10 m/s × 10 s = 100 meters

New Distance Between the Cars

After 10 seconds, Car B has also traveled 100 meters, so the distance between the two cars is:

Distance = 24,900 - 100 = 24,800 meters

Relative Speed After Deceleration

At this point, Car A is still decelerating, and we need to find out how long it will take for Car B to catch up. Car A's speed decreases every second, while Car B maintains a constant speed of 10 m/s. We can find the time it takes for Car B to close the gap using their relative speeds.

Calculating Time to Catch Up

Let’s denote the speed of Car A at any time t as:

v_A(t) = 20 - 2t

The relative speed between Car B and Car A is:

v_rel = 10 - v_A(t) = 10 - (20 - 2t) = 2t - 10

To find the time when Car B catches up, we set the distance covered by Car B equal to the distance Car A has traveled plus the initial distance:

24,800 = 10t + (20t - t² - 100)

Solving this quadratic equation will give us the time it takes for Car B to catch up to Car A.

Final Thoughts

This problem illustrates the dynamics of motion, including the effects of acceleration and deceleration. By breaking it down into manageable steps, we can analyze the situation clearly and arrive at a solution. If you have any further questions or need clarification on any part of this process, feel free to ask!