The position of a particle along x-axis at time t is given by `x=1 + t-t^2`. The distance travelled by the particle in first 2 seconds is

Options

A. 1 m
B. 2 m
C. 2.5 m
D. 3 m

Answer

To find the distance traveled by the particle in the first 2 seconds, we will follow these steps:

### Step 1: Define the position function
The position of the particle along the x-axis is given by the equation:
\[ x(t) = 1 + t - t^2 \]

### Step 2: Find the velocity function
To determine if the particle changes direction, we need to find the velocity, which is the derivative of the position function with respect to time \( t \):
\[ v(t) = \frac{dx}{dt} = \frac{d}{dt}(1 + t - t^2) \]
Calculating the derivative:
\[ v(t) = 0 + 1 - 2t = 1 - 2t \]

### Step 3: Determine when the velocity is zero
Set the velocity function equal to zero to find when the particle changes direction:
\[ 1 - 2t = 0 \]
Solving for \( t \):
\[ 2t = 1 \]
\[ t = \frac{1}{2} \text{ seconds} \]

### Step 4: Calculate the position at key time points
Now we will calculate the position of the particle at \( t = 0 \), \( t = \frac{1}{2} \), and \( t = 2 \) seconds.

- At \( t = 0 \):
\[ x(0) = 1 + 0 - 0^2 = 1 \]

- At \( t = \frac{1}{2} \):
\[ x\left(\frac{1}{2}\right) = 1 + \frac{1}{2} - \left(\frac{1}{2}\right)^2 = 1 + \frac{1}{2} - \frac{1}{4} = 1 + \frac{2}{4} - \frac{1}{4} = 1 + \frac{1}{4} = \frac{5}{4} \]

- At \( t = 2 \):
\[ x(2) = 1 + 2 - 2^2 = 1 + 2 - 4 = -1 \]

### Step 5: Calculate the distance traveled
Now we can calculate the distance traveled in two segments:
1. From \( t = 0 \) to \( t = \frac{1}{2} \):
- The distance is \( |x\left(\frac{1}{2}\right) - x(0)| = \left|\frac{5}{4} - 1\right| = \left|\frac{5}{4} - \frac{4}{4}\right| = \frac{1}{4} \)

2. From \( t = \frac{1}{2} \) to \( t = 2 \):
- The distance is \( |x(2) - x\left(\frac{1}{2}\right)| = |-1 - \frac{5}{4}| = |-1 - 1.25| = |-2.25| = 2.25 \)

### Step 6: Total distance traveled
Now, add the distances from both segments:
\[ \text{Total distance} = \frac{1}{4} + 2.25 = \frac{1}{4} + \frac{9}{4} = \frac{10}{4} = 2.5 \]

### Final Answer
The total distance traveled by the particle in the first 2 seconds is:
\[ \boxed{2.5} \]

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