A car driver going at speed `v` suddenly finds a wide wall at a distance `r` . Should he apply breakes or turn the car in a circle of radius `r` to avoid hitting the wall.

Explanation

To determine whether the car driver should apply brakes or turn the car in a circle of radius \( r \) to avoid hitting the wall, we can analyze the situation step by step.

### Step 1: Understand the scenario
The car is moving at a speed \( v \) and suddenly encounters a wall at a distance \( r \). The driver has two options: apply the brakes to stop the car or turn the car in a circular path with a radius \( r \).

**Hint:** Visualize the scenario to understand the options available to the driver.

### Step 2: Calculate the deceleration needed to stop the car
Using the equation of motion, we can find the deceleration required to stop the car before hitting the wall. The equation we use is:
\[ v^2 = u^2 + 2as \]
Where:
- \( v \) is the final velocity (0 when the car stops),
- \( u \) is the initial velocity (\( v \)),
- \( a \) is the acceleration (or deceleration in this case),
- \( s \) is the distance to the wall (\( r \)).

Rearranging the equation gives:
\[ 0 = v^2 + 2ar \implies a = -\frac{v^2}{2r} \]

**Hint:** Remember that the acceleration is negative because it is deceleration.

### Step 3: Calculate the centripetal acceleration required to turn the car
If the driver decides to turn the car in a circular path of radius \( r \), the required centripetal acceleration \( a_c \) is given by:
\[ a_c = \frac{v^2}{r} \]

**Hint:** Centripetal acceleration is necessary for any object moving in a circular path.

### Step 4: Compare the accelerations
Now, we compare the deceleration required to stop the car with the centripetal acceleration required to turn the car:
- Deceleration when braking: \( a_b = -\frac{v^2}{2r} \)
- Centripetal acceleration when turning: \( a_c = \frac{v^2}{r} \)

To avoid hitting the wall, the driver should ensure that the required centripetal acceleration does not exceed the deceleration available when braking.

### Step 5: Analyze the comparison
From the comparison:
\[ a_b = -\frac{v^2}{2r} \quad \text{and} \quad a_c = \frac{v^2}{r} \]
We can see that:
\[ |a_b| = \frac{v^2}{2r} < a_c = \frac{v^2}{r} \]
This indicates that the braking force is less than the centripetal force required to turn the car.

### Conclusion
Since the required centripetal acceleration to turn the car is greater than the deceleration available when braking, the driver should apply the brakes to avoid hitting the wall.

**Final Answer:** The driver should apply brakes to avoid hitting the wall.

---