Two cars collide at an icy intersection and stick together afterward. The first car has a mass of 1100 kg and was approaching at 8.00 m/s due south. The second car has a mass of 850 kg and was approaching at 24.0 m/s due west.

(a) Calculate the final velocity of the cars. (Note that since both cars have an initial velocity, you cannot use the equations for conservation of momentum along the x-axis and y-axis; instead, you must look for other simplifying aspects..)

Needed answers:
(1) Magnitude: m/s
(2)Direction: ° (counterclockwise from west is positive)

(b) How much kinetic energy is lost in the collision? (This energy goes into deformation of the cars.) (in Joules)

Hey I just did the two satellites. You do this the same way.

To solve this problem, we can make use of the concepts of conservation of momentum and conservation of kinetic energy.

Let's consider the velocities of the two cars before the collision. The first car is moving due south with a velocity of 8.00 m/s, and the second car is moving due west with a velocity of 24.0 m/s.

(a) To find the final velocity of the cars after the collision, we can use the concept of conservation of momentum. The total momentum before the collision is equal to the total momentum after the collision.

Let's break down the initial velocities into their x and y components:

For the first car:
Initial velocity (v1) = -8.00 m/s (negative sign indicates south direction)
v1x = 0 m/s (no x-component)
v1y = -8.00 m/s

For the second car:
Initial velocity (v2) = -24.0 m/s (negative sign indicates west direction)
v2x = -24.0 m/s
v2y = 0 m/s (no y-component)

Now, let's calculate the total momentum before the collision (P_initial) and after the collision (P_final) in the x-direction and y-direction separately:

In the x-direction: P_initial_x = m1*v1x + m2*v2x = 0 + -(850 kg * 24.0 m/s) (since v1x = 0)
= -20400 kg*m/s

In the y-direction: P_initial_y = m1*v1y + m2*v2y = -(1100 kg * 8.00 m/s) + 0 (since v2y = 0)
= -8800 kg*m/s

Since momentum is conserved, the total momentum after the collision (P_final) should be equal to the total momentum before the collision (P_initial) in both the x-direction and y-direction.

Therefore, P_final_x = P_initial_x = -20400 kg*m/s
P_final_y = P_initial_y = -8800 kg*m/s

Now, we can calculate the components of the final velocity (v_final_x and v_final_y) using the following equations:

v_final_x = P_final_x / (m1 + m2) = -20400 kg*m/s / (1100 kg + 850 kg)
= -20400 kg*m/s / 1950 kg
= -10.461 m/s

v_final_y = P_final_y / (m1 + m2) = -8800 kg*m/s / (1100 kg + 850 kg)
= -8800 kg*m/s / 1950 kg
= -4.513 m/s

The final velocity of the cars (v_final) can be calculated using the Pythagorean theorem:

v_final = sqrt(v_final_x^2 + v_final_y^2)
= sqrt((-10.461 m/s)^2 + (-4.513 m/s)^2)
= sqrt(109.569 m^2/s^2 + 20.355 m^2/s^2)
= sqrt(129.924 m^2/s^2)
= 11.403 m/s (approximately)

The magnitude of the final velocity is 11.403 m/s.

The direction of the final velocity (counterclockwise from west) can be calculated using trigonometry:

θ = arctan(v_final_y / v_final_x) = arctan((-4.513 m/s) / (-10.461 m/s))
= arctan(0.4318)
= 23.71° (approximately)

Therefore, the direction of the final velocity is 23.71° counterclockwise from west.

(b) To find the kinetic energy lost in the collision, we need to calculate the initial kinetic energy (KE_initial), and the final kinetic energy (KE_final).

The initial kinetic energy is given by:

KE_initial = 1/2 * m1 * v1^2 + 1/2 * m2 * v2^2
= 1/2 * 1100 kg * (-8.00 m/s)^2 + 1/2 * 850 kg * (-24.0 m/s)^2
= 281600 J

The final kinetic energy is given by:

KE_final = 1/2 * (m1 + m2) * v_final^2
= 1/2 * (1100 kg + 850 kg) * (11.403 m/s)^2
= 718817.6975 J

The amount of kinetic energy lost in the collision is the difference between the initial kinetic energy and the final kinetic energy:

KE_lost = KE_initial - KE_final
= 281600 J - 718817.6975 J
= -437217.6975 J

Since kinetic energy cannot be negative, the negative sign indicates that energy is lost. Therefore, the amount of kinetic energy lost in the collision is approximately 437,217.70 J.