A body is thrown vertically upward with velocity u. The distance travelled by it in the `7^(th) and 8^(th)` seconds are equal. The displacement in `8^(th)` seconds is equal to (take `g=10m//s^(2)`)

Options

A. 5 m
B. 10 m
C. 2.5 m
D. `(5)/(3)m`

Answer

To solve the problem step-by-step, we will use the formula for displacement during the nth second of motion under uniform acceleration. The formula is:

\[ S_n = u + \frac{a}{2} (2n - 1) \]

where:
- \( S_n \) is the displacement during the nth second,
- \( u \) is the initial velocity,
- \( a \) is the acceleration (which will be negative due to gravity when the body is moving upwards),
- \( n \) is the second for which we want to calculate the displacement.

### Step 1: Calculate the displacement in the 7th second

For the 7th second:
- \( n = 7 \)
- \( a = -g = -10 \, \text{m/s}^2 \) (since the body is moving upwards against gravity)

Substituting these values into the formula:

\[ S_7 = u + \frac{-10}{2} (2 \times 7 - 1) \]

Calculating the expression inside the parentheses:

\[ S_7 = u - 5 \times (14 - 1) = u - 5 \times 13 = u - 65 \]

### Step 2: Calculate the displacement in the 8th second

For the 8th second:
- \( n = 8 \)
- The body is now moving downwards after reaching the maximum height, so \( u \) becomes \(-u\) and \( a = g = 10 \, \text{m/s}^2 \).

Substituting these values into the formula:

\[ S_8 = -u + \frac{10}{2} (2 \times 8 - 1) \]

Calculating the expression inside the parentheses:

\[ S_8 = -u + 5 \times (16 - 1) = -u + 5 \times 15 = -u + 75 \]

### Step 3: Set the distances equal

According to the problem, the distances travelled in the 7th and 8th seconds are equal:

\[ S_7 = S_8 \]

Substituting the expressions we found:

\[ u - 65 = -u + 75 \]

### Step 4: Solve for \( u \)

Rearranging the equation:

\[ u + u = 75 + 65 \]
\[ 2u = 140 \]
\[ u = 70 \, \text{m/s} \]

### Step 5: Calculate the displacement in the 8th second

Now that we have \( u \), we can find \( S_8 \):

\[ S_8 = -u + 75 \]
Substituting \( u = 70 \):

\[ S_8 = -70 + 75 = 5 \, \text{m} \]

### Final Answer

The displacement in the 8th second is \( 5 \, \text{m} \).

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