a body travels for 15 seconds from rest with constant acceleration. if it travels S1 distance in first five second and S2 in second five seconds and next five in 3rd five seconds then prove

s1=1/3*s2=1/5*S3

To tackle this problem, we need to analyze the motion of a body under constant acceleration. When an object starts from rest and accelerates uniformly, we can use the equations of motion to find the distances traveled in different time intervals. Let's break it down step by step.

Understanding the Motion

Given that the body starts from rest, we can denote the initial velocity (u) as 0. The constant acceleration is represented as 'a'. The total time of travel is 15 seconds, and we will look at the distances covered in three intervals of 5 seconds each.

Distance Calculation

The formula for the distance traveled under constant acceleration is:

• S = ut + (1/2)at²

Since the initial velocity (u) is 0, the formula simplifies to:

• S = (1/2)at²

Calculating Distances for Each Interval

Let's calculate the distances for each of the three intervals:

First 5 Seconds (S1)

For the first 5 seconds (t = 5), the distance S1 is:

• S1 = (1/2)a(5)² = (1/2)a(25) = 12.5a

Second 5 Seconds (S2)

For the second 5 seconds (t = 10), we need to find the distance from 5 seconds to 10 seconds. The total distance covered in 10 seconds is:

• S(10) = (1/2)a(10)² = (1/2)a(100) = 50a

Thus, the distance S2 is:

• S2 = S(10) - S(5) = 50a - 12.5a = 37.5a

Third 5 Seconds (S3)

For the third 5 seconds (t = 15), we find the total distance covered in 15 seconds:

• S(15) = (1/2)a(15)² = (1/2)a(225) = 112.5a

Thus, the distance S3 is:

• S3 = S(15) - S(10) = 112.5a - 50a = 62.5a

Establishing the Relationships

Now that we have the distances for each interval, we can express them in terms of ratios:

Finding the Ratios

We have:

• S1 = 12.5a
• S2 = 37.5a
• S3 = 62.5a

Now, let's find the ratios:

Ratio of S1 to S2

To find the ratio S1 to S2:

• S1/S2 = (12.5a)/(37.5a) = 12.5/37.5 = 1/3

Ratio of S2 to S3

Next, for the ratio S2 to S3:

• S2/S3 = (37.5a)/(62.5a) = 37.5/62.5 = 3/5
• Thus, S3 = (5/3)S2

Final Relationships

From our calculations, we have established:

• S1 = (1/3)S2
• S2 = (3/5)S3

Rearranging gives us:

• S1 = (1/3)S2 = (1/5)S3

This proves the required relationships between the distances traveled in each of the three intervals. The key takeaway is that under constant acceleration, the distances covered in equal time intervals are not equal but follow a specific ratio based on the time intervals and the nature of acceleration.