A ball falls freely from rest. The ratio of the distance travelled in first, second, third and fourth second is

Options

A. `4:3:2:1`
B. `7:5:3:1`
C. `1:2:3:4`
D. `1:3:5:7`

Answer

To solve the problem of finding the ratio of the distances traveled by a ball falling freely from rest during the first, second, third, and fourth seconds, we can follow these steps:

### Step 1: Understand the motion
A ball falls freely from rest, which means its initial velocity (u) is 0. The acceleration (a) is due to gravity (g).

### Step 2: Use the distance formula
The distance traveled by an object under uniform acceleration can be calculated using the formula:
\[ s = ut + \frac{1}{2} a t^2 \]
Since the initial velocity \( u = 0 \), the formula simplifies to:
\[ s = \frac{1}{2} g t^2 \]

### Step 3: Calculate distances for each second
1. **Distance traveled in the 1st second (s1)**:
\[ s_1 = \frac{1}{2} g (1^2) = \frac{1}{2} g \]

2. **Distance traveled in the 2nd second (s2)**:
The total distance traveled in 2 seconds is:
\[ s_{total, 2} = \frac{1}{2} g (2^2) = \frac{1}{2} g \cdot 4 = 2g \]
The distance traveled in the 2nd second is:
\[ s_2 = s_{total, 2} - s_{total, 1} = 2g - \frac{1}{2} g = \frac{3}{2} g \]

3. **Distance traveled in the 3rd second (s3)**:
The total distance traveled in 3 seconds is:
\[ s_{total, 3} = \frac{1}{2} g (3^2) = \frac{1}{2} g \cdot 9 = \frac{9}{2} g \]
The distance traveled in the 3rd second is:
\[ s_3 = s_{total, 3} - s_{total, 2} = \frac{9}{2} g - 2g = \frac{9}{2} g - \frac{4}{2} g = \frac{5}{2} g \]

4. **Distance traveled in the 4th second (s4)**:
The total distance traveled in 4 seconds is:
\[ s_{total, 4} = \frac{1}{2} g (4^2) = \frac{1}{2} g \cdot 16 = 8g \]
The distance traveled in the 4th second is:
\[ s_4 = s_{total, 4} - s_{total, 3} = 8g - \frac{9}{2} g = 8g - \frac{9}{2} g = \frac{16}{2} g - \frac{9}{2} g = \frac{7}{2} g \]

### Step 4: Write the distances
Now we have:
- \( s_1 = \frac{1}{2} g \)
- \( s_2 = \frac{3}{2} g \)
- \( s_3 = \frac{5}{2} g \)
- \( s_4 = \frac{7}{2} g \)

### Step 5: Calculate the ratio
The ratio of the distances traveled in the first, second, third, and fourth seconds is:
\[ s_1 : s_2 : s_3 : s_4 = \frac{1}{2} g : \frac{3}{2} g : \frac{5}{2} g : \frac{7}{2} g \]
Since \( \frac{1}{2} g \) is common, we can simplify this to:
\[ 1 : 3 : 5 : 7 \]

### Final Answer
The ratio of the distances traveled in the first, second, third, and fourth seconds is:
\[ \boxed{1 : 3 : 5 : 7} \]