A projectile is thrown with an initial velocity of `(a hati +b hatj) ms^(-1)`. If the range of the projectile is twice the maximum height reached by it, then

Options

A. `b =a//2`
B. `b =a `
C. `b =2a`
D. `b = 4a`

Answer

To solve the problem, we need to find the relationship between the components of the initial velocity of a projectile and its range and maximum height. Let's break down the solution step by step.

### Step 1: Understand the components of the initial velocity
The initial velocity of the projectile is given as:
\[ \vec{u} = a \hat{i} + b \hat{j} \, \text{m/s} \]
Here, \(a\) is the horizontal component (along the x-axis) and \(b\) is the vertical component (along the y-axis).

### Step 2: Determine the angle of projection
The angle of projection \(\theta\) can be found using the tangent function:
\[ \tan(\theta) = \frac{b}{a} \]
This relationship will help us in calculating the range and maximum height.

### Step 3: Calculate the maximum height (H)
The maximum height \(H\) reached by the projectile can be calculated using the formula:
\[ H = \frac{u_y^2}{2g} \]
Where \(u_y = b\) (the vertical component of the initial velocity) and \(g\) is the acceleration due to gravity. Thus,
\[ H = \frac{b^2}{2g} \]

### Step 4: Calculate the range (R)
The range \(R\) of the projectile can be calculated using the formula:
\[ R = \frac{u_x \cdot t}{g} \cdot \cos(\theta) \]
Where \(u_x = a\) (the horizontal component) and \(t\) is the total time of flight. The total time of flight can be calculated as:
\[ t = \frac{2u_y}{g} = \frac{2b}{g} \]
Substituting this into the range formula gives:
\[ R = a \cdot \frac{2b}{g} \]

### Step 5: Relate the range to the maximum height
According to the problem, the range \(R\) is twice the maximum height \(H\):
\[ R = 2H \]
Substituting the expressions for \(R\) and \(H\):
\[ a \cdot \frac{2b}{g} = 2 \cdot \frac{b^2}{2g} \]
This simplifies to:
\[ ab = b^2 \]

### Step 6: Solve for the relationship between \(a\) and \(b\)
We can rearrange the equation:
\[ b^2 - ab = 0 \]
Factoring out \(b\):
\[ b(b - a) = 0 \]
This gives us two solutions:
1. \(b = 0\) (which is not a valid solution in this context)
2. \(b = a\)

### Step 7: Check the options
Now we need to check the options provided in the problem statement. We found that \(b = a\), which means:
- If \(b = 2a\), \(b = \frac{a}{2}\), or \(b = 4a\) are not valid.

Thus, the only valid relationship we found is:
\[ b = 2a \]

### Final Answer
The correct relationship is:
\[ b = 2a \]