If the particle is moving along a straight line given by the relation `x=2-3t +4t^3` where s is in cms., and t in sec. Its average velocity during the third sec is

Options

A. 73 cm/s
B. 80 cm/s
C. 85 cm/s
D. 90 cm/s

Answer

To find the average velocity of the particle during the third second, we will follow these steps:

### Step 1: Understand the motion equation
The position of the particle is given by the equation:
\[ x(t) = 2 - 3t + 4t^3 \]
where \( x \) is in centimeters and \( t \) is in seconds.

### Step 2: Calculate the position at the end of the second second (t = 2 seconds)
We need to find the position of the particle at \( t = 2 \) seconds:
\[ x(2) = 2 - 3(2) + 4(2)^3 \]
Calculating this:
\[ x(2) = 2 - 6 + 4(8) \]
\[ x(2) = 2 - 6 + 32 \]
\[ x(2) = 28 \text{ cm} \]

### Step 3: Calculate the position at the end of the third second (t = 3 seconds)
Next, we find the position of the particle at \( t = 3 \) seconds:
\[ x(3) = 2 - 3(3) + 4(3)^3 \]
Calculating this:
\[ x(3) = 2 - 9 + 4(27) \]
\[ x(3) = 2 - 9 + 108 \]
\[ x(3) = 101 \text{ cm} \]

### Step 4: Calculate the average velocity during the third second
The average velocity (\( v_{avg} \)) during the third second can be calculated using the formula:
\[ v_{avg} = \frac{\Delta x}{\Delta t} \]
where \( \Delta x = x(3) - x(2) \) and \( \Delta t = 3 - 2 \).

Substituting the values we found:
\[ \Delta x = 101 \text{ cm} - 28 \text{ cm} = 73 \text{ cm} \]
\[ \Delta t = 3 \text{ s} - 2 \text{ s} = 1 \text{ s} \]

Now, substituting into the average velocity formula:
\[ v_{avg} = \frac{73 \text{ cm}}{1 \text{ s}} = 73 \text{ cm/s} \]

### Final Answer
The average velocity during the third second is:
\[ v_{avg} = 73 \text{ cm/s} \]

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