The maximum possible acceleration of a train starting from the rest and moving on straight track is 10m/s² and maximum possible retardation is 5m/s².the maximum speed that train can achieve is 70 m/s.minimum time in which the train can complete a journey of 1000m ending at rest is 347/2@sec. Where @ is an integer. Find @

To determine the integer value represented by "@" in the time taken for the train to complete a journey of 1000 meters while ending at rest, we need to analyze the situation using the given parameters of the train's motion.

Parameters Provided

• Maximum Acceleration (a): 10 m/s²
• Maximum Retardation (b): 5 m/s²
• Maximum Speed (V_max): 70 m/s
• Distance (d): 1000 m

Breaking Down the Train's Motion

Since the train starts from rest, accelerates to its maximum speed, and then decelerates to rest, we can break the journey into three phases: acceleration, constant speed (if applicable), and deceleration.

Phase 1: Acceleration

During the acceleration phase, we can use the formula:

V = u + at

Where:

• V: final velocity (70 m/s)
• u: initial velocity (0 m/s)
• a: acceleration (10 m/s²)
• t: time of acceleration

Plugging in the values:

70 = 0 + 10t

So, t = 7 seconds.

Distance Covered During Acceleration

We can calculate the distance covered during this acceleration using the formula:

d = ut + (1/2)at²

Substituting the values:

d = 0 + (1/2)(10)(7²) = 245 meters.

Phase 2: Constant Speed

Now, the remaining distance after acceleration can be computed:

Remaining distance = Total distance - Distance during acceleration

Remaining distance = 1000 m - 245 m = 755 m.

Since the maximum speed is 70 m/s, we can find the time taken to cover this distance:

Time = Distance / Speed = 755 m / 70 m/s = 10.7857 seconds.

Phase 3: Deceleration

Next, we need to find out how long it takes to decelerate from 70 m/s to rest (0 m/s) using:

V = u + bt

Here, u = 70 m/s, V = 0, and b = -5 m/s²:

0 = 70 - 5t

Thus, t = 14 seconds.

Calculating Total Time

Now we can sum up the times for each phase:

Total time = Time during acceleration + Time at maximum speed + Time during deceleration

Total time = 7 seconds + 10.7857 seconds + 14 seconds = 31.7857 seconds.

Expressing Total Time

We need to express this total time in the form of 347/2@ seconds. First, let's convert 31.7857 seconds into a fraction:

31.7857 is approximately 63.5714/2. Now, we can set this equal to:

347/2@ = 63.5714/2.

Eliminating the common factor of 2 gives:

347@ = 63.5714.

Now, solving for @:

@ = 63.5714 / 347.

Calculating this gives approximately @ = 0.183. However, since @ must be an integer, we look for the nearest whole number.

Final Answer

The integer value of @ is 1.