A parachutist after bailing out falls `80 m` without friction. When the parachute opens, it decelerates at `2 m//s^(2)`. He reaches the ground with a speed of `20 m//s`. At what height, did he bail out?

Options

A. 180 m
B. 280 m
C. 380 m
D. 480 m

Explanation

To solve the problem step by step, we need to break down the motion of the parachutist into two parts: the free fall before the parachute opens and the deceleration after the parachute opens.

### Step 1: Analyze the free fall before the parachute opens
The parachutist falls freely for a distance of 80 m without any friction. We can use the equations of motion to find the velocity just before the parachute opens.

Using the equation:
\[ V^2 = U^2 + 2gH \]
where:
- \( V \) = final velocity just before the parachute opens,
- \( U \) = initial velocity (0 m/s, since he bailed out),
- \( g \) = acceleration due to gravity (approximately \( 10 \, m/s^2 \)),
- \( H \) = height fallen (80 m).

Substituting the values:
\[ V^2 = 0 + 2 \times 10 \times 80 \]
\[ V^2 = 1600 \]
\[ V = \sqrt{1600} = 40 \, m/s \]

### Step 2: Analyze the motion after the parachute opens
When the parachute opens, the parachutist decelerates at \( 2 \, m/s^2 \) and reaches the ground with a final speed of \( 20 \, m/s \). We can use the same equation of motion to find the height \( h \) from which the parachute opened.

Using the equation:
\[ V^2 = U^2 + 2a h \]
where:
- \( V = 20 \, m/s \) (final velocity),
- \( U = 40 \, m/s \) (initial velocity after free fall),
- \( a = -2 \, m/s^2 \) (deceleration),
- \( h \) = height fallen after the parachute opens.

Substituting the values:
\[ 20^2 = 40^2 + 2(-2)h \]
\[ 400 = 1600 - 4h \]
Rearranging gives:
\[ 4h = 1600 - 400 \]
\[ 4h = 1200 \]
\[ h = \frac{1200}{4} = 300 \, m \]

### Step 3: Calculate the total height from which he bailed out
The total height from which the parachutist bailed out is the sum of the height fallen before the parachute opened and the height fallen after the parachute opened.

Total height \( H_{total} = h + 80 \)
\[ H_{total} = 300 \, m + 80 \, m = 380 \, m \]

### Final Answer
The height from which the parachutist bailed out is **380 meters**.

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