Two cars are traveling in opposite directions on a two lane road. Car A starts from the origin at t =0 and accelerates (moving due East) from rest at 4 m/s2. At t = 0, Car B is moving due West with a velocity of 15 m/s and is 500 m east of car A but accelerating toward the west at 3 m/s2.

a. When do the cars pass by each other?
b. Where are the cars when they pass by each other?
c. How fast is each car traveling when they pass by each other?

a. Write equations for the locations of cars A and B vs. time. Set them equal and solve for t.

b. Use the t solution from a to solve for location (using either of the location vs t equations).

c. VA = 4t
VB = -15 - 3t

Use those equations and the time t from part a, to get VA and VB. Positive velocity is to the east.

To find the answers to these questions, we can use the equations of motion and kinematics. Let's break it down step by step:

a. When do the cars pass by each other?

To find when the cars pass by each other, we need to determine the time it takes for both cars to cover the distance between them.

Let's first calculate the time it takes for Car A to reach the point where the cars meet:

We know the initial velocity (u) of Car A is 0 m/s, and the acceleration (a) is 4 m/s^2. Using the equation v = u + at, where v is the final velocity and t is the time, we can solve for t:

v = u + at
v = 0 + 4t
t = v / 4

Now, let's calculate the time it takes for Car B to reach the point where the cars meet:

Car B is moving towards the west, so its velocity (v) is negative. Given that its initial velocity (u) is -15 m/s and the acceleration (a) is -3 m/s^2, we can use the same equation:

v = u + at
0 = -15 + (-3)t
3t = 15
t = 15 / 3
t = 5

Therefore, the cars pass by each other at t = 5 seconds.

b. Where are the cars when they pass by each other?

To determine the position of the cars when they pass each other, we can use the equation of motion s = ut + (1/2)at^2, where s is the distance, u is the initial velocity, a is the acceleration, and t is the time.

For Car A:
s = 0(5) + (1/2)(4)(5^2)
s = 0 + 40
s = 40 meters (East of the origin)

For Car B:
s = -15(5) + (1/2)(-3)(5^2)
s = -75 - (1/2)(15)(25)
s = -75 - 187.5
s = -262.5 meters (West of the origin)

Therefore, the cars pass by each other at a point 40 meters East of the origin on the road.

c. How fast is each car traveling when they pass by each other?

To find the velocity of each car when they pass by, we can use the equation v = u + at.

For Car A:
v = 0 + 4(5)
v = 0 + 20
v = 20 m/s (East)

For Car B:
v = -15 + (-3)(5)
v = -15 - 15
v = -30 m/s (West)

Therefore, Car A is traveling at 20 m/s towards the East, and Car B is traveling at 30 m/s towards the West when they pass by each other.