Two cars adjacent to each other on a highway. The first car accelerates uniformly at 3m/s2 the moment the light turns green. The second car is moving at 18m/s and is beside the first car at the instant the light changes and continues driving with a constant velocity. When are the cars side by side again?

Write equations for position from the light (X) vs time (t) and set them equal to each other.

Solve for t.

X1 = (a/2) t^2 = 1.5 t^2
X2 = 18 t

18t = 1.5 t^2

1.5 t (t - 12) = 0

One answer is t=0. What is the other?

To find out when the two cars are side by side again, we need to determine the time it takes for the second car traveling at a constant velocity to catch up to the first car, which is accelerating uniformly.

Let's break down the problem step by step:

Step 1: Find the time it takes for the first car to reach the speed of the second car.

The first car is accelerating uniformly at 3 m/s^2. We need to find the time it takes for the first car to reach a speed of 18 m/s, which is the initial speed of the second car.

Using the formula for uniformly accelerated motion: v = u + at, where:
- v is the final velocity (18 m/s in this case),
- u is the initial velocity (0 m/s for the first car),
- a is the acceleration (3 m/s^2), and
- t is the time.

We can rearrange the formula to solve for time (t): t = (v - u) / a.

Substituting the given values, we have t = (18 m/s - 0 m/s) / 3 m/s^2 = 6 seconds.

Thus, it takes the first car 6 seconds to reach the speed of the second car.

Step 2: Calculate the distance covered by the first car during this time.

We know that the first car is accelerating uniformly, so we can use the formula: s = ut + (1/2)at^2, where:
- s is the distance,
- u is the initial velocity, which is 0 m/s,
- t is the time, which is 6 seconds, and
- a is the acceleration, which is 3 m/s^2.

Substituting the given values, we find: s = 0 m/s * 6 s + (1/2) * 3 m/s^2 * (6 s)^2 = 54 m.

Therefore, the first car covers a distance of 54 meters in 6 seconds.

Step 3: Determine when the two cars will be side by side again.

Since the second car is moving at a constant velocity of 18 m/s, it will cover a distance equal to the distance covered by the first car in the same time period. In other words, it needs to cover 54 meters to reach the same position as the first car.

Using the formula s = vt, where:
- s is the distance (54 m),
- v is the velocity (18 m/s), and
- t is the time we need to find.

Rearranging the formula, we find: t = s / v = 54 m / 18 m/s = 3 seconds.

Hence, the two cars will be side by side again after 3 seconds.

Therefore, the answer to the question "When are the cars side by side again?" is 3 seconds.