The acceleration of a particle, moving in a straight line, at time t is `(2t+1) m//sec^(2)` . If `4 m//"sec"` is the initial velocity of the particle then its velocity after 2 sec. is

Options

A. `10 cm//"sec"`
B. `8 cm//"sec"`
C. `4 cm//"sec"`
D. None of these

Answer

To find the velocity of the particle after 2 seconds, we need to integrate the acceleration function to obtain the velocity function. Here’s the step-by-step solution:

### Step 1: Write down the acceleration function
The acceleration \( a(t) \) of the particle is given by:
\[ a(t) = 2t + 1 \quad \text{(in m/s}^2\text{)} \]

### Step 2: Integrate the acceleration to find the velocity
To find the velocity \( v(t) \), we integrate the acceleration function with respect to time \( t \):
\[ v(t) = \int a(t) \, dt = \int (2t + 1) \, dt \]

### Step 3: Perform the integration
Carrying out the integration:
\[ v(t) = \int (2t + 1) \, dt = 2 \cdot \frac{t^2}{2} + t + C = t^2 + t + C \]
where \( C \) is the constant of integration.

### Step 4: Determine the constant of integration
We know the initial velocity \( v(0) = 4 \, \text{m/s} \). Plugging \( t = 0 \) into the velocity equation:
\[ v(0) = 0^2 + 0 + C = C \]
Thus, \( C = 4 \).

### Step 5: Write the complete velocity function
Now we can write the complete velocity function:
\[ v(t) = t^2 + t + 4 \]

### Step 6: Calculate the velocity after 2 seconds
To find the velocity after 2 seconds, substitute \( t = 2 \) into the velocity function:
\[ v(2) = 2^2 + 2 + 4 = 4 + 2 + 4 = 10 \, \text{m/s} \]

### Final Answer
The velocity of the particle after 2 seconds is:
\[ \boxed{10 \, \text{m/s}} \]

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