From the top of a cliff of height 19.6m a ball is projected horizontally. Find the time taken by the ball to reach the ground `(g =9.8 ms^(-2))` .

Answer

To solve the problem of finding the time taken by a ball projected horizontally from a height of 19.6 meters to reach the ground, we can use the equations of motion. Here's a step-by-step solution:

### Step 1: Identify the parameters
- Height of the cliff (h) = 19.6 m
- Acceleration due to gravity (g) = 9.8 m/s²
- Initial vertical velocity (u_y) = 0 m/s (since the ball is projected horizontally)

### Step 2: Use the kinematic equation for vertical motion
The vertical motion of the ball can be described by the following kinematic equation:
\[ s_y = u_y t + \frac{1}{2} g t^2 \]
Where:
- \( s_y \) is the vertical displacement (which will be -19.6 m since the ball is falling down),
- \( u_y \) is the initial vertical velocity (0 m/s),
- \( g \) is the acceleration due to gravity (9.8 m/s²),
- \( t \) is the time taken to reach the ground.

### Step 3: Substitute the known values into the equation
Since the ball is falling down, we can write:
\[ -19.6 = 0 \cdot t + \frac{1}{2} (-9.8) t^2 \]
This simplifies to:
\[ -19.6 = -\frac{9.8}{2} t^2 \]

### Step 4: Simplify the equation
We can simplify the equation further:
\[ -19.6 = -4.9 t^2 \]
Now, we can remove the negative sign from both sides:
\[ 19.6 = 4.9 t^2 \]

### Step 5: Solve for \( t^2 \)
To find \( t^2 \), we divide both sides by 4.9:
\[ t^2 = \frac{19.6}{4.9} \]
Calculating the right side gives:
\[ t^2 = 4 \]

### Step 6: Find \( t \)
Now, take the square root of both sides to find \( t \):
\[ t = \sqrt{4} \]
Thus,
\[ t = 2 \text{ seconds} \]

### Conclusion
The time taken by the ball to reach the ground is **2 seconds**.

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