A 1000-kg car moving east at 80 km/h collides head-on with a 1500-kg car moving west at km/h, and the two cars stick together. (a) Which way does the wreckage move and with what initial speed? (b) How much KE is lost in the collision?

a) To find out which way the wreckage moves and with what initial speed, we can use the principle of conservation of momentum. The total momentum before the collision must be equal to the total momentum after the collision.

Before collision:

1000 kg car has momentum = (mass) * (speed)
= 1000 kg * 80 km/h
= 80000 kg km/h (east)

1500 kg car has momentum = (mass) * (speed)
= 1500 kg * 35 km/h
= 52500 kg km/h (west)

Because the directions are opposite, we can treat them as negative, so the net momentum before the collision is:

80000 kg km/h (east) - 52500 kg km/h (west)
= 27500 kg km/h (east)

After collision (since the two cars stick together, the total mass is 1000 kg + 1500 kg = 2500 kg):

Let V be the velocity of the wreckage after the collision. Then, the momentum after the collision is:
Momentum = (mass) * (velocity) = 2500 kg * V

By the conservation of momentum, we have:

27500 kg km/h (east) = 2500 kg * V

Now, we can solve for V:

V = 27500 kg km/h / 2500 kg = 11 km/h (east)

The wreckage moves towards the east with an initial speed of 11 km/h.

b) To find the kinetic energy (KE) lost in the collision, we can first find the KE before and after the collision and then find the difference between them.

Before collision:

KE of 1000 kg car = (1/2) * (mass) * (speed)^2
= 0.5 * 1000 kg * (80 km/h)^2
= 0.5 * 1000 * 6400 kg (km/h)^2
= 3200000 kg (km/h)^2

KE of 1500 kg car = (1/2) * (mass) * (speed)^2
= 0.5 * 1500 kg * (35 km/h)^2
= 0.5 * 1500 * 1225 kg (km/h)^2
= 918750 kg (km/h)^2

Total KE before collision = 3200000 kg (km/h)^2 + 918750 kg (km/h)^2 = 4118750 kg (km/h)^2

After collision:

Total mass = 2500 kg
Speed = 11 km/h

Total KE after collision = (1/2) * (mass) * (speed)^2
= 0.5 * 2500 kg * (11 km/h)^2
= 0.5 * 2500 * 121 kg (km/h)^2
= 151250 kg (km/h)^2

Now, we can find the KE lost in the collision:

KE lost = KE before collision - KE after collision
= 4118750 kg (km/h)^2 - 151250 kg (km/h)^2
= 3967500 kg (km/h)^2

Therefore, the kinetic energy lost in the collision is 3,967,500 kg (km/h)^2.

To solve this problem, we can use the principle of conservation of momentum and the equation for kinetic energy.

(a) To determine the direction and initial speed of the wreckage, we need to calculate the momentum of the two cars before the collision and after the collision.

The initial momentum of the first car moving east (car A) can be calculated as:

Momentum_A = mass_A * velocity_A

Momentum_A = 1000 kg * 80 km/h

To calculate the momentum of the second car moving west (car B), we need to convert the velocity from km/h to m/s:

velocity_B = - km/h (since it is moving in the opposite direction)
velocity_B = - * (1000 m/1 km) / (3600 s/1 h)
velocity_B = - m/s

Momentum_B = mass_B * velocity_B

Momentum_B = 1500 kg * (- m/s)

Now, let's calculate the total momentum before the collision:

Total momentum before collision = Momentum_A + Momentum_B

(b) To find the total kinetic energy before the collision, we can use the equation:

KE = 1/2 * mass * velocity^2

The kinetic energy of car A is:

KE_A = 1/2 * mass_A * velocity_A^2

KE_A = 1/2 * 1000 kg * (80 km/h)^2

To calculate the kinetic energy of car B, we need to use the velocity in m/s:

KE_B = 1/2 * mass_B * velocity_B^2

KE_B = 1/2 * 1500 kg * (- m/s)^2

Now, let's calculate the total kinetic energy before the collision:

Total KE before collision = KE_A + KE_B

To calculate the total momentum after the collision, we need to consider that the two cars stick together. This means they have the same final velocity. Let's assume the final velocity, V_f, is moving in the east direction. We can now use the conservation of momentum:

Total momentum after collision = Total momentum before collision

Final momentum = (mass_A + mass_B) * V_f

Now we can equate these two equations:

Total momentum before collision = Final momentum

Momentum_A + Momentum_B = (mass_A + mass_B) * V_f

Substituting the values, we can solve for the final velocity:

(1000 kg * 80 km/h) + (1500 kg * (- km/h)) = (1000 kg + 1500 kg) * V_f

Finally, we have the final velocity of the wreckage, and from there we can calculate the initial speed and the direction it moved in after the collision.

To find the amount of kinetic energy lost in the collision, we subtract the total kinetic energy after the collision from the total kinetic energy before the collision:

KE_lost = Total KE before collision - Total KE after collision

To answer these questions, we need to apply the principles of conservation of momentum and kinetic energy.

Let's calculate the initial momentum of each car:

For the 1000-kg car moving east:
Initial momentum = mass * velocity
= 1000 kg * 80 km/h

Since momentum is a vector quantity, and we'll assume eastward velocity as positive, the momentum of the 1000-kg car is:

Momentum1 = 1000 kg * 80 km/h = 80,000 kg*km/h [Eastward]

Similarly, for the 1500-kg car moving west:
Initial momentum = mass * velocity
= 1500 kg * (-50 km/h)

Since velocity is westward, it will be considered negative. Therefore, the momentum of the 1500-kg car is:

Momentum2 = 1500 kg * (-50 km/h) = -75,000 kg*km/h [Westward]

Now, let's find the total momentum of the system after the collision using the conservation of momentum principle, which states that the total momentum before and after a collision remains constant.

Total momentum before collision = Total momentum after collision

Since the two cars stick together, the resulting wreckage will move in the direction of the car with the greater momentum. In this case, the 1000-kg car had a greater initial momentum, so the wreckage will move eastward.

Therefore, we can write:

Momentum1 + Momentum2 (before the collision) = Total momentum (after the collision)

80,000 kg*km/h + (-75,000 kg*km/h) = Total momentum

Total momentum = 5,000 kg*km/h [Eastward]

Next, let's calculate the initial speed of the wreckage. We'll use the concept of conservation of kinetic energy, which states that the total kinetic energy before and after a collision remains constant.

The formula for calculating kinetic energy is given by:

Kinetic energy = 0.5 * mass * velocity^2

The total initial kinetic energy of the system is given by:

Initial kinetic energy = 0.5 * (mass1 * velocity1^2 + mass2 * velocity2^2)

For the 1000-kg car moving east:
Initial kinetic energy1 = 0.5 * 1000 kg * (80 km/h)^2

For the 1500-kg car moving west:
Initial kinetic energy2 = 0.5 * 1500 kg * (50 km/h)^2

Total initial kinetic energy = Initial kinetic energy1 + Initial kinetic energy2

Now, let's calculate the kinetic energy lost in the collision, which is the difference between the total initial kinetic energy and the total final kinetic energy (which is zero because the wreckage comes to rest).

Kinetic energy lost in the collision = Total initial kinetic energy - Total final kinetic energy

Finally, we have all the information we need to answer your questions:

(a) The wreckage moves eastward with an initial speed of 5,000 km/h.
(b) The kinetic energy lost in the collision can be calculated once we determine the total initial kinetic energy and subtract the total final kinetic energy.