A 20.0 g toy car moving to the right at 21.0 cm/s has a head-on collision with a 34.0 g toy car moving in the opposite direction at 47.0 cm/s. If they begin 396.0 cm apart, how long does it take the cars to collide?

To solve this problem, we can use the principles of conservation of momentum and the equation:

(m1 * v1) + (m2 * v2) = (m1 * v1') + (m2 * v2'),

where m1 and m2 are the masses of the cars, v1 and v2 are their initial velocities, and v1' and v2' are their final velocities after the collision.

First, let's convert the given velocities to meters per second (m/s):
v1 = 21.0 cm/s = 0.21 m/s (to the right)
v2 = 47.0 cm/s = -0.47 m/s (to the left), since it's moving in the opposite direction.

Now, let's substitute the given values into the equation and solve for v1' and v2':
(0.020 kg * 0.21 m/s) + (0.034 kg * -0.47 m/s) = (0.020 kg * v1') + (0.034 kg * v2')

Simplifying the equation:
(0.0042 kg·m/s) - (0.01598 kg·m/s) = (0.020 kg * v1') + (0.034 kg * v2')
(-0.01178 kg·m/s) = (0.020 kg * v1') + (0.034 kg * v2')

Since both cars will come to rest after the collision (v1' = 0 m/s and v2' = 0 m/s), we can rewrite the equation as:
(-0.01178 kg·m/s) = (0.020 kg * 0) + (0.034 kg * 0)

Now, we can solve for the unknown values:
(-0.01178 kg·m/s) = 0

As the equation is not valid, it means that the given initial velocities and masses do not result in a head-on collision. However, if there were a collision, we would determine the time it takes for the cars to collide by dividing the initial distance (396.0 cm) between them by the sum of their initial velocities:
t = distance / (v1 + v2)
t = 396.0 cm / (0.21 m/s + (-0.47 m/s))
t = 396.0 cm / (-0.26 m/s)
t = -1523.08 cm/s

Therefore, if there were a head-on collision, it would take approximately -1523.08 seconds for the cars to collide. However, since the velocity is negative, it indicates that the cars would actually separate instead of collide.

To find out how long it takes for the cars to collide, we need to calculate the time it takes for the two cars to cover the initial distance between them.

Let's start by converting the mass of the first car to kilograms:
Mass of the first car = 20.0 g = 0.020 kg

Now let's convert the speed of the first car to meters per second:
Speed of the first car = 21.0 cm/s = 0.21 m/s

Next, let's convert the mass of the second car to kilograms:
Mass of the second car = 34.0 g = 0.034 kg

Now let's convert the speed of the second car to meters per second:
Speed of the second car = 47.0 cm/s = 0.47 m/s

The initial distance between the cars is given as 396.0 cm. Let's convert it to meters:
Initial distance = 396.0 cm = 3.96 m

To find the time it takes for the cars to collide, we can use the following equation:

Time = Distance / Relative speed

The relative speed of the cars can be calculated by subtracting the speeds:

Relative speed = Speed of the first car - Speed of the second car

Let's calculate the relative speed:
Relative speed = 0.21 m/s - (-0.47 m/s) = 0.21 m/s + 0.47 m/s = 0.68 m/s

Now we can calculate the time it takes for the cars to collide:
Time = Initial distance / Relative speed
= 3.96 m / 0.68 m/s
≈ 5.82 s

Therefore, it takes approximately 5.82 seconds for the cars to collide.