Question

Two spheres are projected at angles \(30^{\circ}\) and \(45^{\circ}\) with the horizontal. The maximum height reached by both is same. The ratio of their initial velocities is, \((sin45^{\circ} = \frac{1}{\sqrt{2}},sin30^{\circ} = 0.5)\)

• A. \(2:3\)
• B. \(\sqrt{2}:1\)
• C. \(3:1\)
• D. \(\sqrt{2}:\sqrt{3}\)

Hint:

Maximum height depends on the vertical part of the speed, u sin(theta).

Answer 1

The Correct Option is B

Step 1: Approach
Use the vertical motion only.

Step 2: Vertical motion
The upward speed is $u\sin\theta$, and it falls to zero at the top: $0=(u\sin\theta)^2-2gH$. So $H$ is fixed by $u\sin\theta$.

Step 3: Equal heights
$u_1(0.5)=u_2\left(\dfrac1{\sqrt2}\right)$, so $\dfrac{u_1}{u_2}=\dfrac{2}{\sqrt2}=\sqrt2$.

Step 4: Answer
The ratio is $\sqrt2:1$, option (B).

Final Answer:
Equal heights need equal vertical speeds, so u1 : u2 = sin45 : sin30 = root 2 : 1, option (B). \[ \boxed{\sqrt2:1} \]