A body starts from rest and travels a distance x with uniform acceleration, then it travels a distance 2x with uniform speed, finally it travels a distance 3x with uniform retardation and comes to rest. If the complete motion of the particle is along a straight line, then the ratio of its average velocity to maximum velocity is2/53/54/56/7

The total distance travelled by the body is x (first part) + 2x (second part) + 3x (third part) = 6x.

Let's denote:

• u as the initial velocity (which is 0 because the body starts from rest),
• v as the final velocity (which is also 0 because the body comes to rest),
• a as the acceleration,
• t1 as the time taken in the first part,
• t2 as the time taken in the second part,
• t3 as the time taken in the third part.

From the first equation of motion, v = u + at, we know that the final velocity at the end of the first part is the same as the initial velocity of the second part. Let's denote this velocity as v1.

From the second equation of motion, s = ut + 1/2at^2, we can find the time taken in the first part: x = 0t1 + 1/2a*t1^2, which gives us t1 = sqrt(2x/a).

In the second part, the body travels with uniform speed, so the time taken is t2 = 2x/v1.

In the third part, the body decelerates uniformly to rest, so using the first equation of motion again, but with a negative acceleration -a (because it's deceleration), we get 0 = v1 - a*t3, which gives us t3 = v1/a.

The total time of the journey is t = t1 + t2 + t3.

The average velocity is the total distance divided by the total time, Vavg = 6x/t.

The maximum velocity is v1, which is reached at the end of the first part.

So the ratio of the average velocity to the maximum velocity is Vavg/v1 = (6x/t) / v1.

Without the values of x, a, and v1, we can't calculate a numerical value for this ratio. However, we can say that the ratio will depend on the values of these parameters. If the acceleration a is very large, then the time t1 will be small, the velocity v1 will be large, and the ratio will be small. Conversely, if the acceleration a is very small, then the time t1 will be large, the velocity v1 will be small, and the ratio will be large.