A 1405-kg car moving north at 27.0 m/s is struck by a 2165-kg car moving east at 10.0 m/s. The cars are stuck together. How fast and in what direction do they move immediately after the collision?

Write two equations of conservation of momentum in both the north-south (y) and east-west (x) directions. There will be two unknowns: the x and y components of the final velocity. Each equation can be solved for one final velocity component. (The combined final mass is 3570 kg)

Get the direction from the ratio of the two final velocity components.

We will be glad to critique your work.

To determine the final velocity and direction of the cars after the collision, we can apply the principles of conservation of momentum and combine them using vector addition.

1. Conservation of momentum: In the absence of external forces, the total momentum before the collision is equal to the total momentum after the collision.

Total momentum before collision = Total momentum after collision

The total momentum of the cars before the collision can be calculated by finding the vector sum of individual momenta using the formula:

Total momentum before collision = Momentum of Car 1 + Momentum of Car 2

The momentum of an object is given by the product of its mass and velocity:

Momentum = mass × velocity

2. After the collision, the two cars stick together and move as one object. Therefore, we can calculate the final velocity of the combined cars by dividing the total momentum by the total mass:

Final velocity = Total momentum / Total mass

Let's calculate the momenta and total momentum.

For Car 1:
Mass of Car 1 (m1) = 1405 kg
Initial velocity of Car 1 (v1) = 27.0 m/s

For Car 2:
Mass of Car 2 (m2) = 2165 kg
Initial velocity of Car 2 (v2) = 10.0 m/s

Momentum of Car 1 (p1) = m1 × v1
Momentum of Car 2 (p2) = m2 × v2

Total momentum before collision = p1 + p2

Now, let's calculate the total momentum:

p1 = 1405 kg × 27.0 m/s
p2 = 2165 kg × 10.0 m/s

Total momentum before collision = p1 + p2

Next, we calculate the mass and final velocity of the combined cars:

Total mass (m_total) = m1 + m2
Final velocity (v_final) = Total momentum before collision / Total mass

Finally, we can determine the direction of the final velocity using vector addition.

The final velocity is the vector sum of the initial velocities of Car 1 and Car 2:

v1_final = √(v1^2 + v2^2)

The direction of the final velocity can be determined using trigonometry and the angles of the vectors.

With this approach, you should be able to calculate the final speed and direction of the cars after the collision. I recommend plugging in the given values into the formulas and performing the calculations to find the final results.