A body projected up reaches a point A in its path at the end of 4th second and reaches the ground after 5 seconds from the start. The height of A above the ground is `(g=10 "m/s"^2)`

Options

A. 19.6 m
B. 30.6 m
C. 11 m
D. 20 m

Answer

To solve the problem step by step, we need to analyze the motion of the body projected upwards and determine the height of point A above the ground.

### Step 1: Understand the motion
The body is projected upwards and reaches point A at the end of 4 seconds. It then takes a total of 5 seconds to reach the ground. This means it spends 4 seconds going up and 1 second coming down after reaching the highest point.

### Step 2: Determine the time of ascent and descent
- Time to reach point A = 4 seconds
- Total time of flight = 5 seconds
- Time to reach the maximum height = (Total time - Time to reach point A) = 5 seconds - 4 seconds = 1 second

### Step 3: Calculate the initial velocity (u)
At the maximum height, the final velocity (v) is 0. We can use the equation of motion:
\[ v = u - g t \]
Where:
- \( v = 0 \) (at the maximum height)
- \( g = 10 \, \text{m/s}^2 \) (acceleration due to gravity)
- \( t = 2.5 \, \text{seconds} \) (half of the total time of flight)

Substituting the values:
\[ 0 = u - 10 \times 2.5 \]
\[ u = 10 \times 2.5 = 25 \, \text{m/s} \]

### Step 4: Calculate the maximum height (H)
Using the formula for maximum height:
\[ H = \frac{u^2}{2g} \]
Substituting the values:
\[ H = \frac{25^2}{2 \times 10} \]
\[ H = \frac{625}{20} = 31.25 \, \text{m} \]

### Step 5: Calculate the height of point A (x)
The body reaches point A after 4 seconds. The time taken to reach point A from the maximum height is:
\[ t = 4 \, \text{seconds} - 2.5 \, \text{seconds} = 1.5 \, \text{seconds} \]

Using the equation of motion to find the distance fallen from the maximum height to point A:
\[ x = H - \frac{1}{2} g t^2 \]
Substituting the values:
\[ x = 31.25 - \frac{1}{2} \times 10 \times (1.5)^2 \]
\[ x = 31.25 - \frac{1}{2} \times 10 \times 2.25 \]
\[ x = 31.25 - 11.25 \]
\[ x = 20 \, \text{m} \]

### Conclusion
The height of point A above the ground is \( 20 \, \text{m} \).