A projectile is thrown with an initial velocity of `vecv = (phati+qhatj)` m/s. If the range of the projectile is four times the maximum height reached by it, then

Options

A. p = 2q
B. q = 4p
C. q - 2p
D. q = p

Explanation

To solve the problem, we need to establish the relationship between the range and the maximum height of a projectile thrown with an initial velocity given in vector form. Let's break it down step by step.

### Step 1: Identify the Components of Initial Velocity
The initial velocity of the projectile is given as:
\[ \vec{v} = p \hat{i} + q \hat{j} \text{ m/s} \]
Here, \( p \) is the horizontal component (velocity in the x-direction), and \( q \) is the vertical component (velocity in the y-direction).

### Step 2: Formula for Maximum Height
The maximum height \( H \) reached by a projectile is given by the formula:
\[ H = \frac{v_y^2}{2g} \]
Substituting \( v_y = q \):
\[ H = \frac{q^2}{2g} \]

### Step 3: Formula for Range
The range \( R \) of the projectile is given by the formula:
\[ R = \frac{2v_x v_y}{g} \]
Substituting \( v_x = p \) and \( v_y = q \):
\[ R = \frac{2pq}{g} \]

### Step 4: Relationship Between Range and Maximum Height
According to the problem, the range is four times the maximum height:
\[ R = 4H \]
Substituting the expressions for \( R \) and \( H \):
\[ \frac{2pq}{g} = 4 \left(\frac{q^2}{2g}\right) \]

### Step 5: Simplifying the Equation
We can simplify the equation:
\[ \frac{2pq}{g} = \frac{4q^2}{2g} \]
This simplifies to:
\[ \frac{2pq}{g} = \frac{2q^2}{g} \]
Multiplying both sides by \( g \) (assuming \( g \neq 0 \)):
\[ 2pq = 2q^2 \]

### Step 6: Dividing by 2
Dividing both sides by 2:
\[ pq = q^2 \]

### Step 7: Rearranging the Equation
Rearranging gives:
\[ pq - q^2 = 0 \]
Factoring out \( q \):
\[ q(p - q) = 0 \]

### Step 8: Solving the Equation
This gives us two possible solutions:
1. \( q = 0 \) (which is not a valid solution for projectile motion)
2. \( p - q = 0 \) or \( p = q \)

### Conclusion
Thus, the relationship between the components of the initial velocity is:
\[ p = q \]