A stone thrown vertically up with velocity v reaches three points A, B and C with velocities `v,v/2` and `v/4` respectively. Then AB:BC is

Options

A. `1:1`
B. `2:1`
C. `4:1`
D. `1:4`

Answer

To solve the problem of finding the ratio of distances AB:BC when a stone is thrown vertically upward with an initial velocity \( v \) and reaches points A, B, and C with velocities \( v \), \( \frac{v}{2} \), and \( \frac{v}{4} \) respectively, we can follow these steps:

### Step 1: Define the Displacements
Let:
- \( s_1 \) be the displacement at point A,
- \( s_2 \) be the displacement at point B,
- \( s_3 \) be the displacement at point C.

### Step 2: Calculate \( s_1 \)
At point A, the stone is at its maximum height, and its final velocity is \( v \). Using the equation of motion:
\[ v^2 = u^2 - 2gs_1 \]
Where:
- \( u = v \) (initial velocity),
- \( v = 0 \) (final velocity at maximum height).

Substituting the values:
\[ 0 = v^2 - 2gs_1 \implies s_1 = \frac{v^2}{2g} \]

### Step 3: Calculate \( s_2 \)
At point B, the final velocity is \( \frac{v}{2} \). Again using the equation of motion:
\[ \left(\frac{v}{2}\right)^2 = v^2 - 2gs_2 \]
This simplifies to:
\[ \frac{v^2}{4} = v^2 - 2gs_2 \implies 2gs_2 = v^2 - \frac{v^2}{4} = \frac{3v^2}{4} \]
Thus,
\[ s_2 = \frac{3v^2}{8g} \]

### Step 4: Calculate \( s_3 \)
At point C, the final velocity is \( \frac{v}{4} \). Using the equation of motion:
\[ \left(\frac{v}{4}\right)^2 = v^2 - 2gs_3 \]
This simplifies to:
\[ \frac{v^2}{16} = v^2 - 2gs_3 \implies 2gs_3 = v^2 - \frac{v^2}{16} = \frac{15v^2}{16} \]
Thus,
\[ s_3 = \frac{15v^2}{32g} \]

### Step 5: Calculate Distances AB and BC
Now we can find the distances:
- Distance \( AB = s_2 - s_1 \):
\[ AB = \frac{3v^2}{8g} - \frac{v^2}{2g} = \frac{3v^2}{8g} - \frac{4v^2}{8g} = -\frac{v^2}{8g} \]
(We take the absolute value since distance cannot be negative.)
Thus,
\[ AB = \frac{v^2}{8g} \]

- Distance \( BC = s_3 - s_2 \):
\[ BC = \frac{15v^2}{32g} - \frac{3v^2}{8g} = \frac{15v^2}{32g} - \frac{12v^2}{32g} = \frac{3v^2}{32g} \]

### Step 6: Find the Ratio \( AB:BC \)
Now we can find the ratio:
\[ \frac{AB}{BC} = \frac{\frac{v^2}{8g}}{\frac{3v^2}{32g}} = \frac{v^2}{8g} \cdot \frac{32g}{3v^2} = \frac{32}{24} = \frac{4}{3} \]

### Conclusion
Thus, the ratio \( AB:BC \) is \( 4:1 \).