A rocket is fired upward from the earth's surface such that it creates an acceleration of 19.6 m/sec . If after 5 sec its engine is switched off, the maximum height of the rocket from earth's surface would be

Options

A. 2.45 m
B. 490m
C. 980 m
D. 735m

Answer

To solve the problem of finding the maximum height of a rocket fired upward from the Earth's surface with a given acceleration, we can break the solution down into clear steps:

### Step 1: Determine the initial conditions
The rocket is fired with an acceleration \( a = 19.6 \, \text{m/s}^2 \) for a time \( t = 5 \, \text{s} \). The initial velocity \( u = 0 \, \text{m/s} \) since it starts from rest.

### Step 2: Calculate the final velocity after 5 seconds
Using the formula for final velocity:
\[ v = u + at \]
Substituting the known values:
\[ v = 0 + (19.6 \, \text{m/s}^2)(5 \, \text{s}) = 98 \, \text{m/s} \]
So, the velocity of the rocket after 5 seconds is \( 98 \, \text{m/s} \).

### Step 3: Calculate the distance traveled during the first 5 seconds
Using the formula for distance traveled under constant acceleration:
\[ x = ut + \frac{1}{2} a t^2 \]
Substituting the known values:
\[ x = 0 + \frac{1}{2} (19.6 \, \text{m/s}^2)(5 \, \text{s})^2 \]
Calculating:
\[ x = \frac{1}{2} (19.6)(25) = 9.8 \times 25 = 245 \, \text{m} \]
So, the distance traveled during the first 5 seconds is \( 245 \, \text{m} \).

### Step 4: Calculate the maximum height after the engine is turned off
After 5 seconds, the engine is switched off, and the rocket will continue to rise until its velocity becomes zero due to the acceleration of gravity acting downwards (\( g = 9.8 \, \text{m/s}^2 \)).

Using the formula for maximum height reached under uniform deceleration:
\[ v^2 = u^2 - 2gh \]
Where \( v = 0 \) (final velocity at the maximum height), \( u = 98 \, \text{m/s} \) (initial velocity after 5 seconds), and \( g = 9.8 \, \text{m/s}^2 \).

Rearranging for \( h \):
\[ 0 = (98)^2 - 2(9.8)h \]
\[ 2(9.8)h = (98)^2 \]
\[ h = \frac{(98)^2}{2 \times 9.8} = \frac{9604}{19.6} = 490 \, \text{m} \]
So, the additional height gained after the engine is turned off is \( 490 \, \text{m} \).

### Step 5: Calculate the total maximum height from the Earth's surface
The total maximum height \( H \) is the sum of the height during the powered flight and the height gained after the engine is turned off:
\[ H = x + h = 245 \, \text{m} + 490 \, \text{m} = 735 \, \text{m} \]

### Final Answer
The maximum height of the rocket from the Earth's surface is \( 735 \, \text{m} \).

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