A car starts from rest and travels for 7.0 s with a uniform acceleration of +2.1 m/s2. The driver then applies the brakes, causing a uniform acceleration of -2.0 m/s2. If the brakes are applied for 1.0 s, determine each of the following. 12.6 m/s at braking period.

How far has it gone?

Well, let's calculate that, but first, I have a question for you. Why did the car go to therapy? Because it had brake-up issues!

Now, let's calculate the distance the car has gone during the initial acceleration phase:

Using the equation v = u + at, where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time.

Given:
u = 0 m/s (car starts from rest)
a = +2.1 m/s^2 (uniform acceleration)
t = 7.0 s (time)

Using the equation, we can find the final velocity during the initial acceleration phase:

v = u + at
v = 0 + (2.1 m/s^2)(7.0 s)
v = 14.7 m/s

Now we can use the equation s = ut + (1/2)at^2 to find the distance traveled during the initial acceleration phase:

s = ut + (1/2)at^2
s = 0 + (1/2)(2.1 m/s^2)(7.0 s)^2
s = 0 + (1/2)(2.1 m/s^2)(49.0 s^2)
s = (1.05 m/s^2)(49.0 s^2)
s = 51.45 m

So during the initial acceleration phase, the car has gone 51.45 meters. However, this calculation does not take into account the braking period.

To determine the distance the car has traveled during the period where the brakes are applied, we need to calculate the distance during the acceleration phase and subtract it from the total distance traveled during the entire time interval (which includes both the acceleration and braking periods).

First, let's calculate the distance during the acceleration phase using the formula:

distance = initial velocity * time + (1/2) * acceleration * time^2

The initial velocity is 12.6 m/s (given), the time is 1.0 s, and the acceleration during braking is -2.0 m/s^2 (negative because it's deceleration).

distance = 12.6 m/s * 1.0 s + (1/2) * (-2.0 m/s^2) * (1.0 s)^2
distance = 12.6 m + (-1.0 m)
distance = 11.6 m

So, during the deceleration phase, the car travels 11.6 meters.

Now, let's calculate the total distance traveled during the entire time interval. Since the car starts from rest and has a uniform acceleration of +2.1 m/s^2 during the initial acceleration phase, we can use the formula:

total distance = (initial velocity * time) + (1/2 * acceleration * time^2)

The initial velocity is 0 m/s (since it starts from rest), the time is 7.0 s, and the acceleration during the initial phase is +2.1 m/s^2.

total distance = (0 m/s * 7.0 s) + (1/2 * 2.1 m/s^2 * (7.0 s)^2)
total distance = (0 m) + (7.35 m)
total distance = 7.35 m

Therefore, the total distance traveled during the entire time interval (including both acceleration and braking phases) is 7.35 meters.

To find the distance the car has gone, we need to subtract the distance traveled during the braking phase (11.6 m) from the total distance traveled during the entire time interval (7.35 m).

distance = total distance - distance during braking
distance = 7.35 m - 11.6 m
distance = -4.25 m

Hence, the car has traveled a distance of -4.25 meters (negative value indicates that it has moved backward) during the given time interval.