A ball is rolled off along the edge of a horizontal table with velocity `4m//s`. It hits the ground after time `0.4s`. Which of the following are correct?

Options

A. The height of the table is `0.8m`
B. It hits the ground at an angle of `60^(@)` with the vertical
C. It covers a horizontal distance of `1.6m` from the table
D. It hits the ground with vertical velocity `4m//s`

Answer

To solve the problem step by step, we will analyze the motion of the ball that rolls off the edge of a horizontal table.

### Step 1: Determine the height of the table
The ball is rolled off the table with a horizontal velocity of \(4 \, \text{m/s}\) and it takes \(0.4 \, \text{s}\) to hit the ground. We need to find the height of the table, which can be calculated using the formula for vertical motion:

\[ h = u_y t + \frac{1}{2} g t^2 \]

Where:
- \(h\) = height of the table
- \(u_y\) = initial vertical velocity = \(0 \, \text{m/s}\) (since it rolls off horizontally)
- \(g\) = acceleration due to gravity = \(10 \, \text{m/s}^2\)
- \(t\) = time = \(0.4 \, \text{s}\)

Substituting the values:

\[ h = 0 \cdot 0.4 + \frac{1}{2} \cdot 10 \cdot (0.4)^2 \]
\[ h = 0 + \frac{1}{2} \cdot 10 \cdot 0.16 \]
\[ h = 0.8 \, \text{m} \]

### Step 2: Determine the horizontal distance traveled
The horizontal distance \(x\) can be calculated using the formula:

\[ x = u_x \cdot t \]

Where:
- \(u_x\) = horizontal velocity = \(4 \, \text{m/s}\)
- \(t\) = time = \(0.4 \, \text{s}\)

Substituting the values:

\[ x = 4 \cdot 0.4 = 1.6 \, \text{m} \]

### Step 3: Determine the vertical velocity just before hitting the ground
The vertical velocity \(v_y\) just before hitting the ground can be calculated using the formula:

\[ v_y = u_y + g t \]

Where:
- \(u_y\) = initial vertical velocity = \(0 \, \text{m/s}\)
- \(g\) = acceleration due to gravity = \(10 \, \text{m/s}^2\)
- \(t\) = time = \(0.4 \, \text{s}\)

Substituting the values:

\[ v_y = 0 + 10 \cdot 0.4 = 4 \, \text{m/s} \]

### Step 4: Determine the angle of impact
To find the angle of impact with the vertical, we can use the horizontal and vertical velocities. The angle \(\theta\) can be calculated using:

\[ \tan(\theta) = \frac{v_y}{u_x} \]

Substituting the values:

\[ \tan(\theta) = \frac{4}{4} = 1 \]

Thus,

\[ \theta = 45^\circ \]

### Summary of Results
1. Height of the table: \(0.8 \, \text{m}\) (Option A is correct)
2. Horizontal distance traveled: \(1.6 \, \text{m}\) (Option C is correct)
3. Vertical velocity just before hitting the ground: \(4 \, \text{m/s}\) (Option D is correct)
4. Angle of impact: \(45^\circ\) (Option B is incorrect)

### Final Answer
The correct options are A, C, and D.

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