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 step by step, we need to analyze the motion of the particle given by the equation \( x = 4 - 2t + t^2 \).

### Step 1: Differentiate the displacement function to find velocity
The velocity \( v(t) \) of the particle is the first derivative of the displacement \( x(t) \) with respect to time \( t \).

\[ v(t) = \frac{dx}{dt} = \frac{d}{dt}(4 - 2t + t^2) \]

### Step 2: Calculate the derivative
Now we differentiate each term:

- The derivative of \( 4 \) is \( 0 \).
- The derivative of \( -2t \) is \( -2 \).
- The derivative of \( t^2 \) is \( 2t \).

Putting it all together, we get:

\[ v(t) = 0 - 2 + 2t = 2t - 2 \]

### Step 3: Analyze the velocity function
The velocity function \( v(t) = 2t - 2 \) is a linear function of time \( t \).

### Step 4: Determine when the velocity is zero
To find when the particle changes direction, we set the velocity to zero:

\[ 2t - 2 = 0 \]

Solving for \( t \):

\[ 2t = 2 \implies t = 1 \]

### Step 5: Determine the sign of velocity
- For \( t < 1 \), \( v(t) < 0 \) (the particle is moving in the negative direction).
- For \( t = 1 \), \( v(t) = 0 \) (the particle is at rest).
- For \( t > 1 \), \( v(t) > 0 \) (the particle is moving in the positive direction).

### Step 6: Determine the speed
Speed is the absolute value of velocity. Therefore, we can express speed \( s(t) \) as:

\[ s(t) = |v(t)| = |2t - 2| \]

### Step 7: Write the piecewise function for speed
The speed can be expressed as a piecewise function:

\[ s(t) = \begin{cases} -(2t - 2) & \text{if } t < 1 \\ 2t - 2 & \text{if } t \geq 1 \end{cases} \]

This means:
- For \( t < 1 \), \( s(t) = 2 - 2t \)
- For \( t \geq 1 \), \( s(t) = 2t - 2 \)

### Step 8: Graph the speed function
To graph the speed function:
- For \( t < 1 \), the graph will be a decreasing line starting from \( s(0) = 2 \) down to \( s(1) = 0 \).
- For \( t \geq 1 \), the graph will be an increasing line starting from \( s(1) = 0 \) and continuing upwards.

### Conclusion
The speed of the particle varies with time as described by the piecewise function above, and the graph will reflect these changes in speed.