A railroad car of mass 2.7 104 kg moving at 3.50 m/s collides and couples with two coupled railroad cars, each of the same mass as the single car and moving in the same direction at 1.20 m/s.

(a) What is the speed of the three coupled cars after the collision?
m/s

(b) How much kinetic energy is lost in the collision?
J

a. conservation of momentum

intial=final
M*3.5+2M*1.2=3M V
solve for V

b. intial KE=1/2 M 3.5^2+ M 1.2^2
final KE=3/2 M V^2
lost is the difference

3 meters per second

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

(a) Using the principle of conservation of momentum, we can say that the total momentum before the collision is equal to the total momentum after the collision.

Before the collision:
Momentum of the single car = mass of the single car × velocity of the single car
= (2.7 × 10^4 kg) × (3.50 m/s)
= 9.45 × 10^4 kg⋅m/s

Momentum of the coupled cars = (mass of each coupled car) × (velocity of each coupled car)
= (2 × 2.7 × 10^4 kg) × (1.20 m/s)
= 6.48 × 10^4 kg⋅m/s

Total momentum before the collision = Momentum of the single car + Momentum of the coupled cars
= 9.45 × 10^4 kg⋅m/s + 6.48 × 10^4 kg⋅m/s
= 1.80 × 10^5 kg⋅m/s

After the collision, the three coupled cars will now move together as a single unit, so the mass of the system is 3 times the individual car mass. Let's denote the final velocity of this system as Vf.

Total momentum after the collision = Total mass of the system × Velocity of the system after the collision
= (3 × 2.7 × 10^4 kg) × Vf

Since the total momentum before and after the collision are equal:
1.80 × 10^5 kg⋅m/s = (3 × 2.7 × 10^4 kg) × Vf

Simplifying the equation:
Vf = (1.80 × 10^5 kg⋅m/s) / (3 × 2.7 × 10^4 kg)
Vf = 2.22 m/s

Therefore, the speed of the three coupled cars after the collision is 2.22 m/s.

(b) To find the kinetic energy lost in the collision, we need to compare the total kinetic energy before and after the collision.

Kinetic energy before the collision:

Kinetic energy of the single car = (1/2) × mass of the single car × (velocity of the single car)^2
= (1/2) × (2.7 × 10^4 kg) × (3.50 m/s)^2
= 1.15 × 10^5 J

Kinetic energy of the coupled cars = (1/2) × (mass of each coupled car) × (velocity of each coupled car)^2
= (1/2) × (2 × 2.7 × 10^4 kg) × (1.20 m/s)^2
= 7.78 × 10^4 J

Total kinetic energy before the collision = Kinetic energy of the single car + Kinetic energy of the coupled cars
= 1.15 × 10^5 J + 7.78 × 10^4 J
= 1.93 × 10^5 J

After the collision, all the kinetic energy is lost, as the cars are coupled and moving with the same final velocity. Therefore, the kinetic energy after the collision is zero.

Kinetic energy lost in the collision = Total kinetic energy before the collision - Total kinetic energy after the collision
= 1.93 × 10^5 J - 0 J
= 1.93 × 10^5 J

Therefore, the kinetic energy lost in the collision is 1.93 × 10^5 J.

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

(a) First, let's calculate the initial momentum before the collision. The momentum (p) of an object is defined as the product of its mass (m) and velocity (v):
p = m * v

For the single car:
p1 = (2.7 * 10^4 kg) * (3.50 m/s)

For the coupled cars:
p2 = (2 * 2.7 * 10^4 kg) * (1.20 m/s)

The total initial momentum before the collision is the sum of these two momenta:
p_initial = p1 + p2

Next, we can use the principle of conservation of momentum, which states that the total momentum before the collision is equal to the total momentum after the collision:
p_initial = p_final

Let's assume the final speed of the three coupled cars after the collision is v_final.

For the single car:
p1_final = (2.7 * 10^4 kg) * (v_final)

For the coupled cars:
p2_final = (2 * 2.7 * 10^4 kg) * (v_final)

The total final momentum after the collision is the sum of these two momenta:
p_final = p1_final + p2_final

Using the principle of conservation of momentum, we can equate the initial and final momentum:
p_initial = p_final
p1 + p2 = p1_final + p2_final

Simplifying the equation, we have:
p1 + p2 = (2.7 * 10^4 kg) * (v_final) + (2 * 2.7 * 10^4 kg) * (v_final)

Now, we can solve for the final velocity (v_final) by rearranging the equation:
v_final = (p1 + p2) / [(2.7 * 10^4 kg) + (2 * 2.7 * 10^4 kg)]

Substituting the given values into the equation, we can calculate the final velocity of the three coupled cars:
v_final = (p1 + p2) / (2.7 * 10^4 kg) * (1 + 2)

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

The initial kinetic energy (KE_initial) is the sum of the kinetic energies of the single car and the coupled cars:
KE_initial = (1/2) * m1 * v1^2 + (1/2) * m2 * v2^2

For the single car:
KE1_initial = (1/2) * (2.7 * 10^4 kg) * (3.50 m/s)^2

For the coupled cars:
KE2_initial = (1/2) * (2 * 2.7 * 10^4 kg) * (1.20 m/s)^2

The total initial kinetic energy is the sum of these two kinetic energies:
KE_initial = KE1_initial + KE2_initial

After the collision, the final kinetic energy (KE_final) is the sum of the kinetic energies of the three coupled cars:
KE_final = (1/2) * (3 * 2.7 * 10^4 kg) * (v_final)^2

To find the kinetic energy lost in the collision, we subtract the final kinetic energy from the initial kinetic energy:
Kinetic energy lost = KE_initial - KE_final

Now you can calculate the values for (a) the speed of the three coupled cars after the collision and (b) the amount of kinetic energy lost in the collision using the given values and the equations provided.