Velocity of a projectile at height `15 m` from ground is `v=(20hati+10hatj)m//s`. Here `hati` is in horizontal direction and `hatj` is vertically upwards. Then
Maximum height from a ground is ……..`m`

Options

A. 30
B. 60
C. 40
D. 20

Answer

To find the maximum height of a projectile given its velocity at a height of 15 m, we can follow these steps:

### Step 1: Understand the given velocity
The velocity of the projectile at a height of 15 m is given as:
\[ \vec{v} = 20 \hat{i} + 10 \hat{j} \, \text{m/s} \]
Here, \( \hat{i} \) represents the horizontal component and \( \hat{j} \) represents the vertical component of the velocity.

### Step 2: Identify the vertical component of velocity
From the given velocity, the vertical component of the velocity \( v_y \) at height 15 m is:
\[ v_y = 10 \, \text{m/s} \]

### Step 3: Use the kinematic equation
We can use the kinematic equation that relates the initial vertical velocity \( u_y \), final vertical velocity \( v_y \), acceleration due to gravity \( g \), and the height \( y \):
\[ v_y^2 = u_y^2 + 2a_y y \]
Where:
- \( v_y = 0 \) m/s at the maximum height (the vertical velocity becomes zero at the peak)
- \( a_y = -g = -10 \, \text{m/s}^2 \) (acceleration due to gravity)
- \( y = 15 \, \text{m} \) (the height at which we know the velocity)

### Step 4: Rearranging the equation
Rearranging the equation for \( u_y \):
\[ 0 = u_y^2 - 2g \cdot 15 \]
\[ u_y^2 = 2g \cdot 15 \]

### Step 5: Substitute the values
Substituting \( g = 10 \, \text{m/s}^2 \):
\[ u_y^2 = 2 \cdot 10 \cdot 15 \]
\[ u_y^2 = 300 \]
\[ u_y = \sqrt{300} = 10\sqrt{3} \, \text{m/s} \]

### Step 6: Calculate the maximum height
Now, we can calculate the maximum height \( H \) using the formula:
\[ H = \frac{u_y^2}{2g} \]
Substituting the values:
\[ H = \frac{300}{2 \cdot 10} \]
\[ H = \frac{300}{20} \]
\[ H = 15 \, \text{m} \]

### Step 7: Total maximum height from the ground
The total maximum height from the ground is the height at which we calculated the velocity plus the additional height gained:
\[ \text{Total maximum height} = 15 \, \text{m} + 15 \, \text{m} = 30 \, \text{m} \]

### Final Answer
The maximum height from the ground is:
\[ \text{Maximum height} = 30 \, \text{m} \]

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