Understanding the problem

We are given that a body is moving with uniform acceleration and its speed is initially u. The speed is then doubled while covering a distance S. We are asked to determine the speed of the body when it covers an additional distance S.

Solution

Let's break down the problem into smaller steps to find the solution.

Step 1: Initial speed and distance

Given that the initial speed of the body is u and it covers a distance S, we can use the equation of motion:

\[S = ut + \frac{1}{2}at^2\]

where S is the distance covered, u is the initial speed, a is the uniform acceleration, and t is the time taken.

Since the body is moving with uniform acceleration, we can rewrite the equation as:

\[S = ut + \frac{1}{2}(2a)t^2\]

Simplifying the equation:

\[S = ut + at^2\]

Step 2: Doubling the speed

We are told that the speed of the body is doubled while covering the distance S. Let's assume the new speed after doubling is 2u.

To find the time taken to cover the distance S with the new speed, we can use the equation of motion:

\[S = (2u)t' + \frac{1}{2}a(t')^2\]

where t' is the new time taken.

Again, since the body is moving with uniform acceleration, we can rewrite the equation as:

\[S = (2u)t' + a(t')^2\]

Step 3: Finding the new speed

Now, we need to find the speed of the body when it covers an additional distance S.

Let's assume the final speed after covering the additional distance is v.

Using the equation of motion, we can write:

\[2S = vt + \frac{1}{2}a(t)^2\]

Since the speed is doubled, we can substitute v with 2u:

\[2S = (2u)t + \frac{1}{2}a(t)^2\]

Simplifying the equation:

\[2S = 2ut + \frac{1}{2}a(t)^2\]

We know that the distance covered in the second scenario is an additional S. Therefore, we can substitute 2S with 3S:

\[3S = 2ut + \frac{1}{2}a(t)^2\]

Since the time taken to cover the distance S in the second scenario is the same as the time taken to cover the distance S in the first scenario (t), we can rewrite the equation as:

\[3S = 2u(t) + \frac{1}{2}a(t)^2\]

Simplifying the equation:

\[3S = 2ut + \frac{1}{2}at^2\]

We can recognize that this equation is the same as the equation we derived in Step 1:

\[3S = S\]

Therefore, the speed of the body when it covers an additional distance S would become **3u**.

Conclusion

The speed of the body, initially u, becomes 3u when it covers an additional distance S. This can be derived by using the equations of motion and

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