3 After 4 seconds, the car in Problem # 1 applies the brakes and comes to a full stop after 2 seconds. What is the acceleration of the car and what is the braking distance?

To find the acceleration of the car and the braking distance after it applies the brakes, we can break this problem down into a few logical steps. We’ll use some basic physics equations related to motion. Let's start by identifying the information we have and what we need to find.

Given Information

• Initial velocity (v_i) after 4 seconds: This is the velocity the car reaches before braking.
• Time to stop (t): 2 seconds.
• Final velocity (v_f): 0 m/s (since the car comes to a full stop).

Step 1: Calculate Initial Velocity

Before the car applies the brakes, we need to determine its initial velocity after 4 seconds. If we assume the car was moving at a constant speed during that time, we can use the formula:

v_i = a × t

However, since we don't have the acceleration during the first 4 seconds, let's assume it was moving at a constant speed. If we denote the acceleration during the first 4 seconds as 'a', we can express the initial velocity as:

v_i = a × 4

Step 2: Calculate Acceleration During Braking

Now, we can find the acceleration while the car is braking. We can use the formula:

a = (v_f - v_i) / t

Substituting the known values:

a = (0 - v_i) / 2

Since we need to express this in terms of 'a', we can substitute 'v_i' from our earlier calculation:

a = - (a × 4) / 2

This simplifies to:

a = -2a

This indicates that the acceleration due to braking is negative, which is expected as it is a deceleration. To find the actual value of 'a', we can set it to a specific value, say 'a = 4 m/s²' for the sake of calculation. Thus, the braking acceleration would be:

a = -4 m/s²

Step 3: Calculate Braking Distance

To find the braking distance, we can use the formula:

d = v_i × t + 0.5 × a × t²

Substituting the values we have:

d = (4 m/s) × (2 s) + 0.5 × (-4 m/s²) × (2 s)²

This simplifies to:

d = 8 m - 8 m = 0 m

However, this indicates that we need to ensure our initial velocity is correctly calculated. If we assume the car was traveling at a constant speed of 4 m/s before braking, then:

d = (4 m/s) × (2 s) + 0.5 × (-4 m/s²) × (4 s) = 8 m - 16 m = -8 m

This negative distance indicates an error in our assumptions. Let's clarify: if the car was initially traveling at a higher speed, say 8 m/s, then:

d = (8 m/s) × (2 s) + 0.5 × (-4 m/s²) × (4 s) = 16 m - 16 m = 0 m

Thus, the braking distance depends on the initial speed before braking. If we assume a constant speed of 8 m/s, the braking distance would be 8 meters.

Final Summary

In summary, the acceleration of the car while braking is -4 m/s², and the braking distance will depend on the initial speed before braking. If we assume an initial speed of 8 m/s, the braking distance would be 8 meters. Always ensure to clarify the initial conditions to accurately calculate these values!