Assertion If a particle is projected vertically upwards with velocity u, the maximum height attained by the particle is `h_(1)`. The same particle is projected at angle `30^(@)` from horizontal with the same speed u. Now the maximum height is `h_(2)`. Thus `h_(1) = 4h_(2)`.
Reason In first case, `v =0` at highest point and in second case `v ne 0` at highest point.

Options

A. If both Asseration and Reason are correct and Reason is the correct explanation of Assertion
B. If both Assertion and Reason are correct but Reason is not the correct explanation of Assertion
C. If Assertion is true but Reason is false
D. If Assertion is false but Reason is true

Explanation

To solve the problem, we need to analyze the two scenarios described in the assertion:

1. A particle is projected vertically upwards with an initial velocity \( u \).
2. The same particle is projected at an angle of \( 30^\circ \) with the same initial velocity \( u \).

### Step 1: Calculate the maximum height \( h_1 \) for vertical projection.

When a particle is projected vertically upwards, the maximum height \( h_1 \) can be calculated using the formula:

\[ h_1 = \frac{u^2}{2g} \]

where:
- \( u \) is the initial velocity,
- \( g \) is the acceleration due to gravity.

### Step 2: Calculate the maximum height \( h_2 \) for projection at an angle of \( 30^\circ \).

For a projectile launched at an angle \( \theta \), the vertical component of the initial velocity is given by \( u \sin \theta \). Therefore, when the angle is \( 30^\circ \):

\[ \sin 30^\circ = \frac{1}{2} \]

Thus, the vertical component of the velocity is:

\[ u_y = u \sin 30^\circ = u \cdot \frac{1}{2} = \frac{u}{2} \]

Now, we can calculate the maximum height \( h_2 \) using the formula:

\[ h_2 = \frac{(u_y)^2}{2g} = \frac{\left(\frac{u}{2}\right)^2}{2g} = \frac{u^2/4}{2g} = \frac{u^2}{8g} \]

### Step 3: Relate \( h_1 \) and \( h_2 \).

Now, we can express \( h_1 \) in terms of \( h_2 \):

From our previous calculations:

\[ h_1 = \frac{u^2}{2g} \]
\[ h_2 = \frac{u^2}{8g} \]

To find the relationship between \( h_1 \) and \( h_2 \):

\[ h_1 = 4h_2 \]

This confirms the assertion that \( h_1 = 4h_2 \).

### Step 4: Analyze the reason provided.

The reason states that in the first case, the velocity \( v = 0 \) at the highest point, while in the second case, \( v \neq 0 \) at the highest point.

- In the vertical projection, the particle's velocity becomes zero at the maximum height.
- In the angled projection, while the vertical component of the velocity is zero at the maximum height, the horizontal component remains non-zero.

Thus, the reason is true, but it does not correctly explain why \( h_1 = 4h_2 \). The relationship between the heights is derived from the components of the initial velocity, not from the velocities at the highest point.

### Conclusion

Both the assertion and the reason are true, but the reason is not the correct explanation for the assertion.

### Final Answer

The assertion is true: \( h_1 = 4h_2 \). The reason is also true, but it does not explain the assertion correctly.

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