A body moves in a straight line along x-axis, its distance from the origin is given by the equation `x = 8t-3t ^(2).` The average velocity in the interval from `t =0` to `t =4` is

Options

A. `2m//s`
B. `-16m//s`
C. `-4m//s`
D. `5m//s`

Answer

To find the average velocity of a body moving along the x-axis, we can use the formula for average velocity, which is given by:

\[ \text{Average Velocity} = \frac{\text{Total Displacement}}{\text{Total Time}} \]

### Step 1: Determine the displacement at \( t = 0 \)

Using the equation of motion given:

\[ x(t) = 8t - 3t^2 \]

Substituting \( t = 0 \):

\[ x(0) = 8(0) - 3(0)^2 = 0 \]

### Step 2: Determine the displacement at \( t = 4 \)

Now, substituting \( t = 4 \):

\[ x(4) = 8(4) - 3(4)^2 \]

Calculating this step-by-step:

1. \( 8(4) = 32 \)
2. \( 3(4^2) = 3(16) = 48 \)

Thus,

\[ x(4) = 32 - 48 = -16 \]

### Step 3: Calculate the total displacement

The total displacement from \( t = 0 \) to \( t = 4 \) is:

\[ \text{Displacement} = x(4) - x(0) = -16 - 0 = -16 \text{ meters} \]

### Step 4: Calculate the total time

The total time interval is:

\[ \text{Total Time} = 4 - 0 = 4 \text{ seconds} \]

### Step 5: Calculate the average velocity

Now, substituting the values into the average velocity formula:

\[ \text{Average Velocity} = \frac{\text{Total Displacement}}{\text{Total Time}} = \frac{-16 \text{ meters}}{4 \text{ seconds}} = -4 \text{ m/s} \]

### Final Answer

The average velocity in the interval from \( t = 0 \) to \( t = 4 \) is:

\[ \text{Average Velocity} = -4 \text{ m/s} \]

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