A rocket of initial mass 6000kg ejects mass at constant rate of 200 kg/sec. with constant relative speed of 800 m/sec. The acceleration of the rocket after 5 sec is (neglect gravity)

Options

A. `50 m//s^(2)`
B. `16 m//s^(2)`
C. `60 m//s^(2)`
D. `32 m//s^(2)`

Answer

To solve the problem, we will follow these steps:

### Step 1: Determine the mass of the rocket after 5 seconds
The initial mass of the rocket is given as 6000 kg, and it ejects mass at a constant rate of 200 kg/sec. Therefore, after 5 seconds, the mass ejected will be:

\[ \text{Mass ejected} = \text{Rate of mass ejection} \times \text{Time} = 200 \, \text{kg/sec} \times 5 \, \text{sec} = 1000 \, \text{kg} \]

Now, we can find the mass of the rocket after 5 seconds:

\[ \text{Mass after 5 seconds} = \text{Initial mass} - \text{Mass ejected} = 6000 \, \text{kg} - 1000 \, \text{kg} = 5000 \, \text{kg} \]

### Step 2: Calculate the force exerted by the rocket
The rocket ejects mass with a constant relative speed of 800 m/sec. The thrust (force) generated by the rocket can be calculated using the formula:

\[ \text{Force} = \text{Relative speed} \times \frac{dm}{dt} \]

Where \(\frac{dm}{dt}\) is the rate of mass ejection (200 kg/sec). Therefore, the force is:

\[ \text{Force} = 800 \, \text{m/sec} \times 200 \, \text{kg/sec} = 160000 \, \text{N} \]

### Step 3: Apply Newton's second law to find acceleration
According to Newton's second law, the net force acting on an object is equal to the mass of the object multiplied by its acceleration:

\[ F = m \cdot a \]

We can rearrange this to find the acceleration \(a\):

\[ a = \frac{F}{m} \]

Substituting the values we have:

\[ a = \frac{160000 \, \text{N}}{5000 \, \text{kg}} = 32 \, \text{m/sec}^2 \]

### Final Answer
The acceleration of the rocket after 5 seconds is \(32 \, \text{m/sec}^2\).

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