Question

A projectile is thrown with an initial velocity $(a \hat{i}+b \hat{j}) \text{ m/s}$, where $\hat{i}$ and $\hat{j}$ are unit vectors along horizontal and vertical directions respectively. If the range of the projectile is twice the maximum height reached by it, then

• A. $b = 2a$
• B. $b = 4a$
• C. $b = \frac{a}{2}$
• D. $b = a$

Hint:

For any projectile trajectory, there is a handy shortcut relating range and maximum height: $\tan\theta = \frac{4H}{R}$, where $\theta$ is the angle of projection. Here, since $R = 2H$, the formula yields $\tan\theta = \frac{4H}{2H} = 2$. Since $\tan\theta = \frac{u_y}{u_x} = \frac{b}{a}$, we directly get $\frac{b}{a} = 2 \implies b = 2a$.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The problem presents a projectile launched with a velocity vector whose horizontal component is $a$ and vertical component is $b$. We are given that the horizontal range ($R$) is exactly twice the maximum height ($H$), and we need to determine the mathematical relation between $b$ and $a$.

Step 2: Key Formula or Approach:
The maximum height $H$ and horizontal range $R$ of a projectile can be written in terms of its initial velocity components $u_x$ and $u_y$:
$$H = \frac{u_y^2}{2g}$$ $$R = \frac{2u_x u_y}{g}$$ The problem states the conditional constraint:
$$R = 2H$$

Step 3: Detailed Explanation:
From the given velocity vector $\vec{u} = a\hat{i} + b\hat{j}$, we identify:
Horizontal component, $u_x = a$
Vertical component, $u_y = b$
Substitute these components into the expressions for height and range:
$$H = \frac{b^2}{2g}$$ $$R = \frac{2ab}{g}$$ Now, substitute these expressions into the given condition $R = 2H$:
$$\frac{2ab}{g} = 2 \cdot \left(\frac{b^2}{2g}\right)$$ Simplifying both sides by canceling out the 2 on the right-hand side:
$$\frac{2ab}{g} = \frac{b^2}{g}$$ Since $g \neq 0$, we multiply both sides by $g$:
$$2ab = b^2$$ Assuming the vertical velocity component $b \neq 0$ for a valid projectile motion, we can divide both sides by $b$:
$$2a = b \implies b = 2a$$

Step 4: Final Answer:
The correct relationship is $b = 2a$, which corresponds to option (A).