A ball is rolled off the edge of a horizontal table at a speed of `4 m//"second"`. It hits the ground after 0.4 second . Which statement given below is true

Options

A. It hits the ground at a horizontal distance 1.6 m from the edge of the table
B. The speed with which it hits the ground is `4.0 ms^(-1)`
C. Height of the table 1m
D. It hits the ground at an angle of `60^(@)` to the horizontal

Explanation

To solve the problem step by step, we will analyze the motion of the ball as it rolls off the table. The motion can be divided into horizontal motion and vertical motion.

### Step 1: Identify the Horizontal Motion
The ball rolls off the table with a horizontal speed of \( u_x = 4 \, \text{m/s} \). The time it takes to hit the ground is given as \( t = 0.4 \, \text{s} \).

### Step 2: Calculate the Horizontal Distance
To find the horizontal distance \( d \) the ball travels before hitting the ground, we can use the formula:
\[ d = u_x \cdot t \]
Substituting the known values:
\[ d = 4 \, \text{m/s} \cdot 0.4 \, \text{s} = 1.6 \, \text{m} \]
Thus, the ball hits the ground at a horizontal distance of \( 1.6 \, \text{m} \) from the edge of the table.

### Step 3: Analyze the Vertical Motion
Next, we consider the vertical motion to find the height of the table. The initial vertical velocity \( u_y = 0 \) (since the ball rolls off horizontally), and the acceleration due to gravity \( a = 10 \, \text{m/s}^2 \).

Using the equation of motion for vertical displacement:
\[ h = u_y \cdot t + \frac{1}{2} a t^2 \]
Substituting the known values:
\[ h = 0 \cdot 0.4 + \frac{1}{2} \cdot 10 \cdot (0.4)^2 \]
Calculating:
\[ h = \frac{1}{2} \cdot 10 \cdot 0.16 = 0.8 \, \text{m} \]
So, the height of the table is \( 0.8 \, \text{m} \), which contradicts the statement that the height is \( 1 \, \text{m} \).

### Step 4: Determine the Final Speed at Impact
To find the speed with which it hits the ground, we need to calculate the vertical component of the velocity just before impact. Using the formula:
\[ v_y = u_y + a \cdot t \]
Substituting the known values:
\[ v_y = 0 + 10 \cdot 0.4 = 4 \, \text{m/s} \]
The total speed at impact can be calculated using Pythagorean theorem since the horizontal and vertical motions are independent:
\[ v = \sqrt{(u_x)^2 + (v_y)^2} = \sqrt{(4)^2 + (4)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \, \text{m/s} \]
Thus, the speed with which it hits the ground is not \( 4 \, \text{m/s} \).

### Step 5: Determine the Angle of Impact
The angle \( \theta \) at which the ball hits the ground can be determined using:
\[ \tan \theta = \frac{v_y}{u_x} = \frac{4}{4} = 1 \]
Thus, \( \theta = 45^\circ \), which is not \( 60^\circ \).

### Conclusion
Based on the calculations:
1. The ball hits the ground at a horizontal distance of \( 1.6 \, \text{m} \) from the edge of the table (True).
2. The speed with which it hits the ground is not \( 4 \, \text{m/s} \) (False).
3. The height of the table is \( 0.8 \, \text{m} \) (False).
4. It hits the ground at an angle of \( 45^\circ \) (False).

### Final Answer
The only true statement is: **It hits the ground at a horizontal distance of 1.6 m from the edge of the table.**

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