A stone is thrown vertically upwards. When stone is at a height half of its maximum height, its speed is `10 ms^-1` , then the maximum height attained by the stone is (`g= 10 ms^(-2)`)

Options

A. `8m`
B. `10m`
C. `15m`
D. `20m`

Answer

To solve the problem step by step, we will use the equations of motion and the information provided in the question.

### Step 1: Understand the problem
We have a stone thrown vertically upwards. At half of its maximum height, the speed of the stone is given as \(10 \, \text{m/s}\). We need to find the maximum height attained by the stone, given that the acceleration due to gravity \(g = 10 \, \text{m/s}^2\).

### Step 2: Define variables
Let:
- \( h \) = maximum height attained by the stone
- \( u \) = initial velocity of the stone
- \( V \) = velocity of the stone at height \( \frac{h}{2} \) = \( 10 \, \text{m/s} \)

### Step 3: Use the equation of motion
We can use the equation of motion to relate the initial velocity, final velocity, acceleration, and displacement. The equation is:
\[ V^2 = u^2 + 2a s \]
Where:
- \( V = 10 \, \text{m/s} \) (velocity at height \( \frac{h}{2} \))
- \( a = -g = -10 \, \text{m/s}^2 \) (acceleration due to gravity, acting downwards)
- \( s = \frac{h}{2} \) (displacement from the initial position to half the maximum height)

Substituting the values into the equation:
\[ (10)^2 = u^2 + 2(-10)\left(\frac{h}{2}\right) \]
This simplifies to:
\[ 100 = u^2 - 10h \]
Rearranging gives us:
\[ u^2 = 10h + 100 \tag{1} \]

### Step 4: Relate maximum height to initial velocity
The maximum height \( h \) can also be expressed in terms of the initial velocity using the formula:
\[ h = \frac{u^2}{2g} \]
Substituting \( g = 10 \, \text{m/s}^2 \):
\[ h = \frac{u^2}{20} \tag{2} \]

### Step 5: Substitute equation (1) into equation (2)
Now, we will substitute equation (1) into equation (2):
\[ h = \frac{10h + 100}{20} \]
Multiplying both sides by 20 to eliminate the fraction:
\[ 20h = 10h + 100 \]
Rearranging gives:
\[ 20h - 10h = 100 \]
\[ 10h = 100 \]
Dividing both sides by 10:
\[ h = 10 \, \text{m} \]

### Conclusion
The maximum height attained by the stone is \( \boxed{10 \, \text{m}} \).