Motion in One Dimension: Finding Initial Velocity and Acceleration

This solution explains how to determine the initial velocity and acceleration of a body when the distance travelled in the $n^{th}$ second is provided.

Understanding the Formula for Distance in the $n^{th}$ Second

In kinematics, the distance ($S_n$) covered by a body moving with constant acceleration ($a$) during the $n^{th}$ second is given by a specific formula. This formula relates the distance to the initial velocity ($u$) and the acceleration ($a$). The standard formula is:

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

Here:

• $S_n$ represents the distance traveled in the $n^{th}$ second.
• $u$ represents the initial velocity of the body.
• $a$ represents the constant acceleration of the body.
• $n$ represents the specific second (e.g., 1st second, 2nd second, etc.).

Relating the Given Information to the Standard Formula

The problem states that the distance traveled by the body in the $n^{th}$ second is given by:

$S_n = (7 + 5n) \text{ m}$

To find the initial velocity ($u$) and acceleration ($a$), we need to compare this given expression with the standard formula. First, let's rearrange the standard formula to a more comparable format:

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

Distributing the $\frac{a}{2}$ term:

$S_n = u + a \cdot n - \frac{a}{2}$

Rearranging to group the constant terms and the term with $n$:

$S_n = \left(u - \frac{a}{2}\right) + (a)n$

Step-by-Step Calculation

Now, we compare the rearranged standard formula with the given formula $S_n = 7 + 5n$. We can equate the coefficients of $n$ and the constant terms:

Given Formula: $S_n = 7 + 5n$

Standard Formula: $S_n = \left(u - \frac{a}{2}\right) + a \cdot n$

1. Equating the coefficient of $n$:

   By comparing the terms containing $n$ in both formulas, we get:

   $a = 5$

   This tells us that the acceleration ($a$) of the body is $5 \text{ ms}^{-2}$.

2. Equating the constant terms:

   By comparing the constant terms (terms independent of $n$) in both formulas, we get:

   $u - \frac{a}{2} = 7$

3. Solving for the initial velocity ($u$):

   We already found that $a = 5 \text{ ms}^{-2}$. Substitute this value into the equation from the previous step:

   $u - \frac{5}{2} = 7$

   $u - 2.5 = 7$

   Now, solve for $u$:

   $u = 7 + 2.5$

   $u = 9.5$

   Therefore, the initial velocity ($u$) of the body is $9.5 \text{ ms}^{-1}$.

Conclusion

Based on the comparison and calculations, the initial velocity of the body is $9.5 \text{ ms}^{-1}$ and the acceleration is $5 \text{ ms}^{-2}$.

Summary of Results

The key parameters found are:

• Initial Velocity ($u$): $9.5 \text{ ms}^{-1}$
• Acceleration ($a$): $5 \text{ ms}^{-2}$

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