a car starts and accelerates uniformly at a rate of 2.0 m / s ^ 2 in 6.0 s. Then, it maintains its constant speed for 30 seconds. Finally, the shrinkage slows down uniformly and stops to 5.0 s. find the total distance traveled by the car.

What do you mean by the SHRINKAGE slowing down?

What do you mean by "stops TO 5.0 s" ?
5.0 s from when?

To find the total distance traveled by the car, we can break down the motion into three parts and calculate the distance traveled in each part:

1. Part 1: Acceleration phase (0 to 6.0 s)
During this phase, the car starts from rest and accelerates uniformly at a rate of 2.0 m/s^2 for 6.0 seconds. To find the distance traveled during this phase, we'll use the kinematic equation:

distance = (initial velocity * time) + (0.5 * acceleration * time^2)

The initial velocity is 0 m/s (as the car starts from rest), the time is 6.0 seconds, and the acceleration is 2.0 m/s^2. Plugging these values into the equation:

distance = (0 * 6.0) + (0.5 * 2.0 * 6.0^2)
= 0 + (0.5 * 2.0 * 36)
= 0 + 36.0
= 36.0 meters

2. Part 2: Constant speed phase (6.0 to 36.0 s)
During this phase, the car maintains a constant speed for 30 seconds. Since the speed is constant, the distance traveled is simply the product of the speed and time:

distance = speed * time

The speed is the same as at the end of the acceleration phase, which is given by the equation:

speed = initial velocity + (acceleration * time)

Initial velocity = (acceleration * time) = (2.0 m/s^2)(6.0 s) = 12.0 m/s
Speed = 12.0 m/s

Plugging this value into the distance equation:

distance = 12.0 m/s * 30.0 s
= 360.0 meters

3. Part 3: Deceleration phase (36.0 to 41.0 s)
During this phase, the car slows down uniformly and comes to a stop in 5.0 seconds. To find the distance traveled during this phase, we'll again use the kinematic equation:

distance = (initial velocity * time) + (0.5 * acceleration * time^2)

The initial velocity is the speed at the end of the constant speed phase, which is 12.0 m/s, the time is 5.0 seconds, and the acceleration is calculated as:

acceleration = (final velocity - initial velocity) / time
(Final velocity is 0 m/s since the car comes to a stop)

acceleration = (0 m/s - 12.0 m/s) / 5.0 s
= -12.0 m/s / 5.0 s (deceleration is negative)
= -2.4 m/s^2

Plugging these values into the distance equation:

distance = (12.0 m/s * 5.0 s) + (0.5 * -2.4 m/s^2 * 5.0^2)
= 60.0 m - 30.0 m
= 30.0 meters

Finally, to get the total distance traveled by the car, we add up the distances from the three parts:

Total distance = distance in Part 1 + distance in Part 2 + distance in Part 3
= 36.0 m + 360.0 m + 30.0 m
= 426.0 meters

Therefore, the total distance traveled by the car is 426.0 meters.