Heather is sitting on her car at a spotlight. At the instant light turns green, she starts accelerating at 2.2 m/s/s. At the same moment a truck rumbles past her car going a constant speed of 12.2 m/s. Heather is infuriated at being passed by a truck and she keeps accelerating until she gets ahead of the truck.

A) How much time will it take her to catch up with the truck?
B) How far down the street will she will catch up with the truck?
C) How fast will she be moving at this point?

solve for t when the distances are equal:

0 + 1.1t^2 = 12.2t

Now recall that since she started from a dead stop,

s = 1/2 at^2
v = at

To solve this problem, we can use the kinematic equations of motion. These equations relate the initial velocity (in this case, 0 m/s for both Heather and the truck), the acceleration, the time, and the distance covered.

A) To determine how much time it will take Heather to catch up with the truck, we need to find the time at which their positions are equal. We can use the equation:

distance(Heather) = distance(truck)

Let's denote the time it takes for Heather to catch up with the truck as t.

First, we need to find the distance Heather travels before catching up with the truck. We can use the equation of motion:

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

For Heather:
distance(Heather) = 0 * t + (1/2) * 2.2 * t^2 = 1.1 * t^2

For the truck:
distance(truck) = 12.2 * t

Setting these two equal, we have:
1.1t^2 = 12.2 t

Simplifying, we get:
1.1t^2 - 12.2t = 0

Now, we can factor out t:
t(1.1t - 12.2) = 0

From this equation, we can see that there are two possible solutions: t = 0 (which is the initial time) and t = 12.2/1.1.

Since t = 0 is not a valid solution in this case (as it represents the initial time when the light turns green), we have:
t = 12.2/1.1

Evaluating this, we find:
t ≈ 11.09 seconds

So, it will take Heather approximately 11.09 seconds to catch up with the truck.

B) To determine how far down the street Heather will catch up with the truck, we can use the equation of motion for distance:

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

For Heather:
distance(Heather) = 0 * (12.2/1.1) + (1/2) * 2.2 * (12.2/1.1)^2

Simplifying, we find:
distance(Heather) ≈ 79.11 meters

Therefore, Heather will catch up with the truck approximately 79.11 meters down the street.

C) To determine how fast Heather will be moving at this point, we can use the equation of motion for velocity:

final velocity = initial velocity + acceleration * time

For Heather:
final velocity(Heather) = 0 + 2.2 * (12.2/1.1)

Simplifying, we find:
final velocity(Heather) ≈ 24.13 m/s

Therefore, Heather will be moving at approximately 24.13 m/s when she catches up with the truck.