A truck is stopped at a stoplight. When the light turns green, the truck accelerates at 2.0 m/s2. At the same instant, a car passes the truck going 13 m/s. Where and when does the truck catch up with the car?

_________m (from the intersection)

To find where and when the truck catches up with the car, we need to determine the time it takes for the truck to catch up and then use that time to find the distance from the intersection.

Let's first find the time it takes for the truck to catch up with the car. We can use the equation:

(distance) = (initial velocity) * (time) + (0.5) * (acceleration) * (time)^2

For the car:
(distance) = (car's velocity) * (time)

For the truck:
(distance) = (truck's initial velocity) * (time) + (0.5) * (truck's acceleration) * (time)^2

Since the distance traveled by both vehicles when the truck catches up with the car is the same, we have:

(car's velocity) * (time) = (truck's initial velocity) * (time) + (0.5) * (truck's acceleration) * (time)^2

Now, let's plug in the given values:
(car's velocity) = 13 m/s
(truck's initial velocity) = 0 m/s (since it is stopped)
(truck's acceleration) = 2.0 m/s^2

Using this equation, we can now solve for time:

13 * t = 0 + 0.5 * 2.0 * t^2

Simplifying the equation:

13 * t = t^2

Rearranging the equation:

t^2 - 13 * t = 0

Factoring out t:

t * (t - 13) = 0

Setting each factor to zero:

t = 0 or t - 13 = 0

Since time cannot be zero for this problem, we have:

t - 13 = 0
t = 13

Therefore, it takes 13 seconds for the truck to catch up with the car.

Now, let's find the distance from the intersection where the truck catches up with the car. We can use the equation:

(distance) = (initial velocity) * (time) + (0.5) * (acceleration) * (time)^2

Using the truck's equation:

(distance) = (0 m/s) * (13 s) + (0.5) * (2.0 m/s^2) * (13 s)^2

Simplifying the equation:

(distance) = 0 + (0.5) * (2.0) * (169)

(distance) = 0 + (1) * (169)

(distance) = 169 m

Therefore, the truck catches up with the car at a distance of 169 meters from the intersection.

To determine where and when the truck catches up with the car, we can use kinematic equations and set the displacement of the two vehicles equal to each other.

Let's assume that the truck catches up to the car after a time interval of t seconds and that their distances from the intersection are x (truck) and y (car).

For the truck, we can use the equation:
x = 1/2 * a * t^2 + v0 * t
where,
a = acceleration of the truck = 2.0 m/s^2 (given)
v0 = initial velocity of the truck = 0 m/s (since it was stopped)

For the car, we can use the equation:
y = v1 * t
where,
v1 = velocity of the car = 13 m/s (given)

Since the truck catches up to the car, their displacements are equal at the time they meet. Therefore, we can set x = y.

1/2 * a * t^2 + v0 * t = v1 * t

Simplifying the equation, we have:
1/2 * 2.0 * t^2 = 13 * t
t^2 = 13 * t
t = 13 seconds

Now, we can substitute the value of t back into any of the displacement equations to find the distance from the intersection:

For the truck:
x = 1/2 * a * t^2 + v0 * t
x = 1/2 * 2.0 * (13)^2 + 0 * 13
x = 169 meters

Therefore, the truck catches up with the car at a distance of 169 meters from the intersection.