Question

6. In a non-injury chain-reaction accident, on a foggy freeway, car 1 (mass of 2000

kg) moving at 15.0 m/s to the right elastically collides with car 2, initially at rest.
The mass of car 2 is 1500 kg. In turn, car 2 then goes on to lock bumpers (that is,
it is a completely inelastic collision with car 3, which has a mass of 2500 kg and
was also at rest. Determine the speed of all cars (i.e., car 1, car 2, and the car 2/3
combo) immediately after this unfortunate accident.

Answer 1

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

Given:

Mass of car 1 (m1) = 2000 kg

Initial velocity of car 1 (v1_initial) = 15.0 m/s to the right

Mass of car 2 (m2) = 1500 kg

Mass of car 3 (m3) = 2500 kg

Since the collision between car 1 and car 2 is described as elastic, we can use the conservation of momentum to find the final velocities of car 1 and car 2. The conservation of momentum states that the total momentum before the collision is equal to the total momentum after the collision.

Before the collision:

Initial momentum = m1 * v1_initial + m2 * 0 (since car 2 is initially at rest)

After the collision:

Final momentum = m1 * v1_final + m2 * v2_final

Using the conservation of momentum, we can equate the initial momentum to the final momentum:

m1 * v1_initial + m2 * 0 = m1 * v1_final + m2 * v2_final

Simplifying the equation:

m1 * v1_initial = m1 * v1_final + m2 * v2_final

Since car 2 and car 3 have a completely inelastic collision, they stick together and move as one unit. We can use the conservation of momentum again to find the final velocity of the car 2/3 combo.

Before the collision:

Initial momentum = m2 * v2_final + m3 * 0 (since car 3 is initially at rest)

After the collision:

Final momentum = (m2 + m3) * v_final_combo

Using the conservation of momentum:

m2 * v2_final + m3 * 0 = (m2 + m3) * v_final_combo

Simplifying the equation:

m2 * v2_final = (m2 + m3) * v_final_combo

Now we have a system of two equations with two unknowns (v1_final and v2_final), which we can solve simultaneously.

First, let's solve the first equation for v1_final:

m1 * v1_initial = m1 * v1_final + m2 * v2_final

v1_final = (m1 * v1_initial - m2 * v2_final) / m1

Next, let's solve the second equation for v2_final:

m2 * v2_final = (m2 + m3) * v_final_combo

v2_final = (m2 + m3) * v_final_combo / m2

Now we substitute the expression for v2_final into the expression for v1_final:

v1_final = (m1 * v1_initial - m2 * [(m2 + m3) * v_final_combo / m2]) / m1

v1_final = (m1 * v1_initial - (m2 + m3) * v_final_combo) / m1

Now we substitute the values:

v1_final = (2000 kg * 15.0 m/s - (1500 kg + 2500 kg) * v_final_combo) / 2000 kg

Simplifying the equation:

v1_final = (30000 kg·m/s - 4000 kg * v_final_combo) / 2000 kg

v1_final = (30.0 m/s - 2.0 * v_final_combo)

Now, let's substitute the expression for v2_final into the expression for v_final_combo:

v2_final = (m2 + m3) * v_final_combo / m2

v2_final = (1500 kg + 2500 kg) * v_final_combo / 1500 kg

v2_final = 4000 kg * v_final_combo / 1500 kg

v2_final = 2.67 * v_final_combo

Now we can substitute this expression into the equation for v1_final:

v1_final = (30.0 m/s - 2.0 * (2.67 * v_final_combo))

v1_final = 30.0 m/s - 5.34 * v_final_combo

Now we can solve for v_final_combo by equating the expressions for v1_final and v2_final:

30.0 m/s - 5.34 * v_final_combo = 2.67 * v_final_combo

Combine like terms:

30.0 m/s = 8.01 * v_final_combo

Solving for v_final_combo:

v_final_combo = 30.0 m/s / 8.01

Calculating the value:

v_final_combo ≈ 3.746 m/s

Now we can substitute this value back into the equation for v1_final to find v1_final:

v1_final = 30.0 m/s - 5.34 * v_final_combo

Substituting the value:

v1_final ≈ 30.0 m/s - 5.34 * 3.746 m/s

Calculating the value:

v1_final ≈ 8.99 m/s

Therefore, immediately after the collision, the speeds of the cars are approximately:

Car 1 (v1_final) = 8.99 m/s to the right

Car 2 (v2_final) = 2.67 * v_final_combo ≈ 2.67 * 3.746 m/s ≈ 10.01 m/s to the right

Car 2/3 combo (v_final_combo) = 3.746 m/s to the right

Sorry if my explanation is too long, i hope your understand! :)