A particle moves along x - axis in such a way that its x - co - ordinate varies with time according to the equation `x=4-2t+t^(2)`. The speed of the particle will vary with time as

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Answer

To solve the problem, we need to find the speed of the particle as a function of time given the position function \( x(t) = 4 - 2t + t^2 \).

### Step-by-Step Solution:

1. **Identify the Position Function**:
The position of the particle as a function of time is given by:
\[ x(t) = 4 - 2t + t^2 \]

2. **Differentiate the Position Function**:
To find the speed of the particle, we need to compute the derivative of the position function with respect to time \( t \). This derivative gives us the velocity \( v(t) \):
\[ v(t) = \frac{dx}{dt} = \frac{d}{dt}(4 - 2t + t^2) \]

3. **Calculate the Derivative**:
- The derivative of the constant \( 4 \) is \( 0 \).
- The derivative of \( -2t \) is \( -2 \).
- The derivative of \( t^2 \) is \( 2t \).

Therefore, the velocity function becomes:
\[ v(t) = 0 - 2 + 2t = 2t - 2 \]

4. **Determine the Speed**:
Speed is the absolute value of velocity. Thus, the speed \( s(t) \) is given by:
\[ s(t) = |v(t)| = |2t - 2| \]

5. **Analyze the Speed Function**:
The expression \( |2t - 2| \) indicates that the speed will vary depending on the value of \( t \):
- For \( t < 1 \), \( 2t - 2 < 0 \) so \( s(t) = -(2t - 2) = 2 - 2t \).
- For \( t = 1 \), \( s(t) = 0 \).
- For \( t > 1 \), \( 2t - 2 > 0 \) so \( s(t) = 2t - 2 \).

### Final Expression for Speed:
Thus, the speed of the particle varies with time as:
\[ s(t) = |2t - 2| \]