A ball is rolled oof along the edge of table (horizontal) with velocity 4 m/s . IT hits the ground after time 0.4 s. Which one of the following is wrong ?

Options

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

Answer

To solve the problem, we will analyze the motion of the ball that is rolled off the edge of the table. We will determine the height of the table, the angle at which the ball hits the ground, the horizontal distance traveled, and the final velocity of the ball upon impact.

### Step-by-Step Solution:

1. **Identify the Given Data:**
- Horizontal velocity (u_x) = 4 m/s
- Time of flight (t) = 0.4 s
- Acceleration due to gravity (g) = 10 m/s² (approximate value)

2. **Calculate the Height of the Table:**
We can use the kinematic equation for vertical motion:
\[ s = ut + \frac{1}{2} a t^2 \]
Here, the initial vertical velocity (u_y) = 0, the displacement (s) = -h (height of the table), and acceleration (a) = -g = -10 m/s². Plugging in the values:
\[ -h = 0 \cdot 0.4 + \frac{1}{2} \cdot (-10) \cdot (0.4)^2 \]
\[ -h = -5 \cdot 0.16 \]
\[ -h = -0.8 \text{ m} \]
Thus, the height of the table (h) = 0.8 m.

3. **Calculate the Final Velocity in the Y-Direction:**
Using the equation for final velocity:
\[ v_y = u_y + a_y t \]
\[ v_y = 0 + (-10) \cdot 0.4 \]
\[ v_y = -4 \text{ m/s} \]
The negative sign indicates the direction is downward.

4. **Determine the Angle of Impact:**
The horizontal component of the velocity (v_x) remains constant:
\[ v_x = 4 \text{ m/s} \]
Now, we can find the angle (θ) with respect to the vertical using the tangent function:
\[ \tan(\theta) = \frac{v_x}{|v_y|} = \frac{4}{4} = 1 \]
Therefore,
\[ \theta = 45^\circ \]
This means the ball hits the ground at an angle of 45 degrees with the horizontal, not 60 degrees with the vertical.

5. **Calculate the Horizontal Distance (Range):**
The horizontal distance (R) can be calculated as:
\[ R = u_x \cdot t \]
\[ R = 4 \cdot 0.4 = 1.6 \text{ m} \]

6. **Final Velocity Upon Impact:**
The magnitude of the resultant velocity (v) can be calculated using Pythagorean theorem:
\[ v = \sqrt{v_x^2 + v_y^2} = \sqrt{4^2 + (-4)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \text{ m/s} \]

### Summary of Findings:
- Height of the table: 0.8 m (Correct)
- Angle of impact: 45 degrees (Incorrect statement in the options)
- Horizontal distance: 1.6 m (Correct)
- Final velocity upon impact: 4 m/s in both x and y directions (Correct)

### Conclusion:
The wrong statement among the options provided is that the ball hits the ground at an angle of 60 degrees with the vertical. The correct angle is 45 degrees.