A car starts from rest, accelerate at 2m/s , for 15sec , it then continues at a steady speed for further 25sec and decelerate to rest in 5ecs . Find

the time taken to cover *two-third* of the distance.

plot velocity vs time. first 15 seconds, v will increase steadily, then constant, then down.

The area under the velocity-time curve is distance.

To find the time taken to cover two-thirds of the distance, we need to break down the motion of the car into three parts: acceleration, constant speed, and deceleration.

Let's calculate the distance covered in each segment first:

1. Acceleration:
Using the formula for distance covered during uniform acceleration:
distance = (initial velocity * time) + (0.5 * acceleration * time^2)

Given:
initial velocity (u) = 0 m/s (since the car starts from rest)
acceleration (a) = 2 m/s^2
time (t) = 15 sec

distance_accl = (0 * 15) + (0.5 * 2 * 15^2)
distance_accl = 0 + (0.5 * 2 * 225)
distance_accl = 0 + 225
distance_accl = 225 meters

2. Constant Speed:
Since the car continues at a steady speed, we can use the formula for distance covered during uniform motion:
distance = speed * time

Given:
speed = 2 m/s (as the car is moving at a steady speed)
time = 25 sec

distance_const_speed = 2 * 25
distance_const_speed = 50 meters

3. Deceleration:
Using the formula for distance covered during uniform deceleration:
distance = (final velocity * time) - (0.5 * acceleration * time^2)

Given:
final velocity (v) = 0 m/s (since the car comes to rest)
acceleration (a) = -2 m/s^2 (negative as it is deceleration)
time (t) = 5 sec

distance_decl = (0 * 5) - (0.5 * (-2) * 5^2)
distance_decl = 0 - (0.5 * (-2) * 25)
distance_decl = 0 - (-25)
distance_decl = 25 meters

Now, let's calculate the time taken to cover two-thirds of the total distance.

Total distance covered = distance_accl + distance_const_speed + distance_decl
Total distance = 225 + 50 + 25
Total distance = 300 meters

Two-thirds of the total distance = (2/3) * Total distance
Two-thirds distance = (2/3) * 300
Two-thirds distance = 200 meters

To calculate the time taken to cover two-thirds of the distance, we need to consider only the acceleration and constant speed segments, as the deceleration segment brings the car to rest.

Time taken to cover two-thirds distance = (distance_accl / (distance_accl + distance_const_speed)) * time_accl
Time taken to cover two-thirds distance = (225 / (225 + 50)) * 15
Time taken to cover two-thirds distance = (225 / 275) * 15
Time taken to cover two-thirds distance = 0.81818 * 15
Time taken to cover two-thirds distance ≈ 12.27 seconds

Therefore, the time taken to cover two-thirds of the distance is approximately 12.27 seconds.

To find the time taken to cover two-thirds of the distance, we need to break down the motion of the car into different segments and calculate the distance covered in each segment.

Let's calculate the distances covered in each segment:

1) Acceleration phase:
The car starts from rest and accelerates at 2 m/s^2 for 15 seconds.
We can use the formula for distance covered during uniform acceleration:
distance = (initial velocity * time) + (0.5 * acceleration * time^2)
In this case, the initial velocity is 0 m/s, acceleration is 2 m/s^2, and time is 15 seconds.
So, the distance covered during the acceleration phase is:
distance1 = (0 * 15) + (0.5 * 2 * 15^2)

2) Steady speed phase:
The car continues at a steady speed for 25 seconds, which means it covers a distance equal to the product of its speed and time.
Since the car is moving at a constant speed during this phase, the distance covered is simply:
distance2 = speed * time
But we need to know the speed of the car during this phase. To find that, we need to know the distance covered during the acceleration phase. So, we'll calculate that first.

3) Deceleration phase:
The car decelerates to rest in 5 seconds, so its final velocity is 0 m/s.
Using the same formula for distance covered during uniform acceleration:
distance = (initial velocity * time) + (0.5 * acceleration * time^2)
This time, the initial velocity is the speed of the car during the steady speed phase (which we'll calculate), acceleration is -2 m/s^2 (negative since it's deceleration), and time is 5 seconds.
So, the distance covered during the deceleration phase is:
distance3 = (speed * 5) + (0.5 * -2 * 5^2)

Now, we can calculate the distance covered during the acceleration phase and then use it to find the speed during the steady speed phase:
distance1 = (0 * 15) + (0.5 * 2 * 15^2)
distance1 = 0 + (0.5 * 2 * 225)
distance1 = 0 + 225
distance1 = 225 meters

Now, we can calculate the speed during the steady speed phase:
distance2 = speed * 25
speed = distance2 / 25

Finally, we can calculate the distance covered during the deceleration phase and then calculate the total distance covered:
distance3 = (speed * 5) + (0.5 * -2 * 5^2)
distance3 = (distance2 / 25 * 5) + (-2 * 5^2)
distance3 = (distance2 / 5) - 50

The total distance covered is the sum of all three distances:
total_distance = distance1 + distance2 + distance3

Now, we can find the time taken to cover two-thirds of the distance:
two_thirds_distance = (2/3) * total_distance
time = (two_thirds_distance / speed)