The brakes on your car can slow you at a rate of `5.2m//s^(2)`. (a) If you are going 146 km/h and suddenly see a state trooper, what is the minimum time in which you can get your car under the 90 km/h speed limit ? (The answer reveals the futility of braking to keep your high speed from being detected with a radar or loser gun. ) (b) Graph `x` versus t and v versus t for such a slowing .

Answer

To solve the problem, we will break it down into two parts:

### Part (a): Minimum Time to Slow Down

1. **Convert Speeds to m/s**:
- The initial speed \( u = 146 \, \text{km/h} \).
- The final speed \( v = 90 \, \text{km/h} \).
- To convert km/h to m/s, we use the conversion factor \( \frac{5}{18} \):
\[ u = 146 \times \frac{5}{18} = 40.56 \, \text{m/s} \]
\[ v = 90 \times \frac{5}{18} = 25 \, \text{m/s} \]

2. **Identify Given Values**:
- Initial velocity \( u = 40.56 \, \text{m/s} \)
- Final velocity \( v = 25 \, \text{m/s} \)
- Acceleration (retardation) \( a = -5.2 \, \text{m/s}^2 \)

3. **Use the First Equation of Motion**:
- The first equation of motion is given by:
\[ v = u + at \]
- Rearranging for time \( t \):
\[ t = \frac{v - u}{a} \]

4. **Substituting the Values**:
- Substitute \( v = 25 \, \text{m/s} \), \( u = 40.56 \, \text{m/s} \), and \( a = -5.2 \, \text{m/s}^2 \):
\[ t = \frac{25 - 40.56}{-5.2} \]
\[ t = \frac{-15.56}{-5.2} \approx 2.99 \, \text{s} \]

### Part (b): Graphs of \( x \) vs \( t \) and \( v \) vs \( t \)

1. **Position vs Time Graph (\( x \) vs \( t \))**:
- The equation for displacement is:
\[ x = ut + \frac{1}{2} a t^2 \]
- Substitute \( u = 40.56 \, \text{m/s} \), \( a = -5.2 \, \text{m/s}^2 \):
\[ x = 40.56t - \frac{1}{2} \cdot 5.2 \cdot t^2 \]
\[ x = 40.56t - 2.6t^2 \]
- This is a quadratic equation, and the graph will be a downward-opening parabola.

2. **Velocity vs Time Graph (\( v \) vs \( t \))**:
- The equation for velocity is:
\[ v = u + at \]
- Substitute \( u = 40.56 \, \text{m/s} \) and \( a = -5.2 \, \text{m/s}^2 \):
\[ v = 40.56 - 5.2t \]
- This is a linear equation, and the graph will be a straight line with a negative slope.

### Summary of Results:
- Minimum time to reduce speed from 146 km/h to 90 km/h is approximately **2.99 seconds**.
- The \( x \) vs \( t \) graph is a downward-opening parabola.
- The \( v \) vs \( t \) graph is a straight line with a negative slope.