The position of a body moving along x-axis at time t is given by `x= (t^(2)-4t+6)m`. The distance travelled by body in time interval t = 0 to t = 3 s is

Options

A. 5 m
B. 7 m
C. 4 m
D. 3 m

Answer

To solve the problem of finding the distance traveled by a body moving along the x-axis from time \( t = 0 \) to \( t = 3 \) seconds, given the position function \( x(t) = t^2 - 4t + 6 \), we will follow these steps:

### Step 1: Calculate the position at \( t = 0 \)
Substituting \( t = 0 \) into the position function:
\[ x(0) = (0)^2 - 4(0) + 6 = 6 \text{ m} \]

### Step 2: Calculate the position at \( t = 2 \)
Substituting \( t = 2 \) into the position function:
\[ x(2) = (2)^2 - 4(2) + 6 = 4 - 8 + 6 = 2 \text{ m} \]

### Step 3: Calculate the distance traveled from \( t = 0 \) to \( t = 2 \)
The distance traveled from \( t = 0 \) to \( t = 2 \) is the absolute difference in position:
\[ \text{Distance from } t = 0 \text{ to } t = 2 = |x(2) - x(0)| = |2 - 6| = 4 \text{ m} \]

### Step 4: Calculate the position at \( t = 3 \)
Substituting \( t = 3 \) into the position function:
\[ x(3) = (3)^2 - 4(3) + 6 = 9 - 12 + 6 = 3 \text{ m} \]

### Step 5: Calculate the distance traveled from \( t = 2 \) to \( t = 3 \)
The distance traveled from \( t = 2 \) to \( t = 3 \) is:
\[ \text{Distance from } t = 2 \text{ to } t = 3 = |x(3) - x(2)| = |3 - 2| = 1 \text{ m} \]

### Step 6: Calculate the total distance traveled from \( t = 0 \) to \( t = 3 \)
Adding the distances from both intervals:
\[ \text{Total distance} = \text{Distance from } t = 0 \text{ to } t = 2 + \text{Distance from } t = 2 \text{ to } t = 3 = 4 \text{ m} + 1 \text{ m} = 5 \text{ m} \]

### Final Answer
The total distance traveled by the body from \( t = 0 \) to \( t = 3 \) seconds is \( 5 \text{ m} \).

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