A truck is stopped at a stoplight. When the light turns green, the truck accelerates at 2.2 m/s2. At the same instant, a car passes the truck going 12 m/s. Where and when does the truck catch up with the car?

They go the same distance in the same time.

truck: d=1/2 a t^2
car: d=12t
set the distances equal
12t=1/2 a t^2
t=24/a

To find out where and when the truck catches up with the car, we need to analyze their motion using the concept of relative motion.

Let's assume the position of the truck at the time the traffic light turned green as the starting point and measure the distance from this point. We'll also consider the time when the truck catches up with the car as the reference point.

Now, let's break down the problem into three steps:

Step 1: Determine the equations for the motion of both the truck and the car.
Step 2: Find the time when the truck catches up with the car.
Step 3: Use this time to calculate the distance at which the truck catches up with the car.

Step 1: Determine the equations for the motion of both the truck and the car:
The equation for the motion of the car can be represented as:
x_car = 12t

Here, x_car represents the distance covered by the car, and t represents time.

The equation for the motion of the truck can be represented as:
x_truck = ut + (1/2)at^2

Here, x_truck represents the distance covered by the truck, u represents the initial velocity of the truck (which is zero as it is stopped), a represents the acceleration of the truck (2.2 m/s^2), and t represents time.

Step 2: Find the time when the truck catches up with the car:
We need to find the time (t) when x_truck = x_car.

Substituting the equations for x_car and x_truck, we have:
12t = (1/2)2.2t^2

Simplifying the equation, we get:
2.2t^2 - 12t = 0

Factoring out t, we get:
t(2.2t - 12) = 0

This equation has two solutions - t = 0 (which represents the starting point) and t = 12/2.2 ≈ 5.45 seconds.

Therefore, the truck catches up with the car after approximately 5.45 seconds.

Step 3: Use this time to calculate the distance at which the truck catches up with the car:
Substituting the value of t into one of the equations (let's use x_car = 12t), we can calculate the distance:
x_car = 12(5.45) ≈ 65.40 meters

Therefore, the truck catches up with the car approximately 65.40 meters away from the starting point.

To summarize, the truck catches up with the car after approximately 5.45 seconds and at a distance of approximately 65.40 meters from the starting point.