The following equations give the position `x(t)` of a particle in four situations ( in each equation, `x` is meters, `t` is in seconds, and `t gt 0` ): (1) `x= 3t-2`, (2) `x= - 4t^(2)-2`, (3) `x= 2//t^(2)`, and (4) `x= -2` . (a) In which situation is the velocity v of the particle constant ? (b) In which is v in the negative `x` direction ?

Answer

To solve the problem, we will analyze each of the four equations given for the position \( x(t) \) of a particle and determine the velocity in each case.

### Step-by-Step Solution:

1. **Identify the equations:**
- (1) \( x(t) = 3t - 2 \)
- (2) \( x(t) = -4t^2 - 2 \)
- (3) \( x(t) = \frac{2}{t^2} \)
- (4) \( x(t) = -2 \)

2. **Calculate the velocity for each equation:**
- The velocity \( v(t) \) is given by the derivative of the position function \( x(t) \) with respect to time \( t \), i.e., \( v(t) = \frac{dx}{dt} \).

3. **For equation (1):**
\[ x(t) = 3t - 2 \]
\[ v(t) = \frac{dx}{dt} = 3 \]
- Here, the velocity is constant (3 m/s).

4. **For equation (2):**
\[ x(t) = -4t^2 - 2 \]
\[ v(t) = \frac{dx}{dt} = -8t \]
- The velocity is not constant; it depends on \( t \) and is negative for \( t > 0 \).

5. **For equation (3):**
\[ x(t) = \frac{2}{t^2} \]
- Rewrite as \( x(t) = 2t^{-2} \).
\[ v(t) = \frac{dx}{dt} = -4t^{-3} = -\frac{4}{t^3} \]
- The velocity is not constant; it depends on \( t \) and is negative for \( t > 0 \).

6. **For equation (4):**
\[ x(t) = -2 \]
\[ v(t) = \frac{dx}{dt} = 0 \]
- Here, the velocity is constant (0 m/s).

### Summary of Results:
- **(a)** The situations where the velocity \( v \) is constant are:
- **Equation (1)**: \( v = 3 \) (constant)
- **Equation (4)**: \( v = 0 \) (constant)

- **(b)** The situations where the velocity \( v \) is in the negative \( x \) direction are:
- **Equation (2)**: \( v = -8t \) (negative for \( t > 0 \))
- **Equation (3)**: \( v = -\frac{4}{t^3} \) (negative for \( t > 0 \))

### Final Answers:
- **(a)** Situations with constant velocity: **1 and 4**
- **(b)** Situations with velocity in the negative \( x \) direction: **2 and 3**