A parachutist after bailing out falls 50m without friction. When parachute opens, it decelerates at `2m//s^2`. He reaches the ground with a speed of `3m//s`. At what height, did the bail out?

Options

A. (a) 182m
B. (b) 91m
C. (c) 111m
D. (d) 293m

Explanation

To solve the problem step by step, we will follow these steps:

### Step 1: Understand the motion of the parachutist
The parachutist falls freely for a distance of 50 m before the parachute opens. After the parachute opens, he decelerates at a rate of \(2 \, \text{m/s}^2\) and eventually reaches the ground with a speed of \(3 \, \text{m/s}\).

### Step 2: Use kinematic equations
We can use the kinematic equation to relate the final velocity, initial velocity, acceleration, and distance traveled after the parachute opens. The equation is:

\[ v^2 = u^2 + 2as \]

Where:
- \(v\) = final velocity (3 m/s)
- \(u\) = initial velocity (velocity just before the parachute opens)
- \(a\) = acceleration (which will be negative since it is deceleration, so \(a = -2 \, \text{m/s}^2\))
- \(s\) = distance traveled after the parachute opens (which we need to find)

### Step 3: Find the initial velocity when the parachute opens
To find \(u\), we first need to calculate the speed of the parachutist just before the parachute opens. We can use the following kinematic equation for free fall:

\[ v^2 = u^2 + 2gh \]

Where:
- \(h = 50 \, \text{m}\) (the distance fallen before the parachute opens)
- \(g = 9.8 \, \text{m/s}^2\) (acceleration due to gravity)

Substituting the values, we have:

\[ u^2 = 0 + 2 \cdot 9.8 \cdot 50 \]
\[ u^2 = 980 \]
\[ u = \sqrt{980} \approx 31.3 \, \text{m/s} \]

### Step 4: Substitute values into the kinematic equation
Now we can substitute \(u\) into the kinematic equation we derived earlier:

\[ (3)^2 = (31.3)^2 + 2(-2)s \]
\[ 9 = 979.69 - 4s \]
\[ 4s = 979.69 - 9 \]
\[ 4s = 970.69 \]
\[ s = \frac{970.69}{4} \approx 242.67 \, \text{m} \]

### Step 5: Calculate the total height from which the parachutist bailed out
The total height \(H\) from which the parachutist bailed out is the sum of the distance fallen before the parachute opens and the distance fallen after the parachute opens:

\[ H = 50 + s \approx 50 + 242.67 \approx 292.67 \, \text{m} \]

### Final Answer
The height from which the parachutist bailed out is approximately \(292.67 \, \text{m}\).

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