A particle moves with constant acceleration along a straight line streaming from rest. The percentage increase in its displacement during the 4th second compared to that in the 3rd second is

Options

A. `33%`
B. `40%`
C. `66%`
D. `77%`

Explanation

To solve the problem step by step, we will use the formula for displacement during the nth second for a particle moving with constant acceleration.

### Step 1: Understand the formula for displacement in the nth second
The displacement \( s_n \) during the nth second is given by the formula:
\[ s_n = u + \frac{1}{2} a (2n - 1) \]
where \( u \) is the initial velocity, \( a \) is the acceleration, and \( n \) is the second during which we want to calculate the displacement.

### Step 2: Calculate displacement during the 3rd second
Since the particle starts from rest, we have \( u = 0 \). For \( n = 3 \):
\[ s_3 = 0 + \frac{1}{2} a (2 \times 3 - 1) = \frac{1}{2} a (6 - 1) = \frac{1}{2} a \times 5 = \frac{5}{2} a \]

### Step 3: Calculate displacement during the 4th second
Now, for \( n = 4 \):
\[ s_4 = 0 + \frac{1}{2} a (2 \times 4 - 1) = \frac{1}{2} a (8 - 1) = \frac{1}{2} a \times 7 = \frac{7}{2} a \]

### Step 4: Find the percentage increase in displacement
The percentage increase in displacement from the 3rd second to the 4th second can be calculated using the formula:
\[ \text{Percentage Increase} = \frac{s_4 - s_3}{s_3} \times 100 \]
Substituting the values we found:
\[ \text{Percentage Increase} = \frac{\frac{7}{2} a - \frac{5}{2} a}{\frac{5}{2} a} \times 100 \]

### Step 5: Simplify the expression
Calculating the numerator:
\[ s_4 - s_3 = \frac{7}{2} a - \frac{5}{2} a = \frac{2}{2} a = a \]
Now substituting back:
\[ \text{Percentage Increase} = \frac{a}{\frac{5}{2} a} \times 100 \]
The \( a \) cancels out:
\[ \text{Percentage Increase} = \frac{2}{5} \times 100 = 40\% \]

### Final Answer
The percentage increase in displacement during the 4th second compared to that in the 3rd second is **40%**.

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