An air-track cart with mass m=0.20kg and speed v0=1.5m/s approaches two other carts that are at rest and have masses 2m and 3m, as indicated in (Figure 1) . The carts have bumpers that make all the collisions elastic.

Find the final speed of cart 1, assuming the air track extends indefinitely in either direction.

Find the final speed of cart 2, assuming the air track extends indefinitely in either direction.

Find the final speed of cart 3, assuming the air track extends indefinitely in either direction.

Cart 1: 0.75 m/s

Cart 2: 0.5 m/s
Cart 3: 0 m/s

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

First, let's define the parameters:

m1 = mass of cart 1 = 0.20 kg
m2 = mass of cart 2 = 2m = 2 * 0.20 kg = 0.40 kg
m3 = mass of cart 3 = 3m = 3 * 0.20 kg = 0.60 kg
v1 = initial velocity of cart 1 = 1.5 m/s
v2 = initial velocity of cart 2 = 0 m/s (at rest)
v3 = initial velocity of cart 3 = 0 m/s (at rest)
v1' = final velocity of cart 1
v2' = final velocity of cart 2
v3' = final velocity of cart 3

Using conservation of momentum (total momentum before collision = total momentum after collision), we can write the equation:

m1 * v1 + m2 * v2 + m3 * v3 = m1 * v1' + m2 * v2' + m3 * v3'

Since v2 = 0 m/s and v3 = 0 m/s, the equation simplifies to:

m1 * v1 = m1 * v1' + m2 * v2' + m3 * v3'

Now, let's use conservation of kinetic energy (total kinetic energy before collision = total kinetic energy after collision):

(1/2) * m1 * v1^2 + (1/2) * m2 * v2^2 + (1/2) * m3 * v3^2 = (1/2) * m1 * v1'^2 + (1/2) * m2 * v2'^2 + (1/2) * m3 * v3'^2

Since v2 = 0 m/s and v3 = 0 m/s, the equation simplifies to:

(1/2) * m1 * v1^2 = (1/2) * m1 * v1'^2 + (1/2) * m2 * v2'^2 + (1/2) * m3 * v3'^2

Now we have two equations with two unknowns (v1' and v2'). We can solve these equations simultaneously using the given values.

First, let's find v1':

m1 * v1 = m1 * v1' + m2 * v2' + m3 * v3'
0.20 kg * 1.5 m/s = 0.20 kg * v1' + 0.40 kg * v2' + 0.60 kg * v3' (substituting the known values)
0.30 kg m/s = 0.2 kg v1' + 0.4 kg v2' + 0.6 kg v3' (simplifying)

Next, let's find v2' using the conservation of kinetic energy equation:

(1/2) * m1 * v1^2 = (1/2) * m1 * v1'^2 + (1/2) * m2 * v2'^2 + (1/2) * m3 * v3'^2
(1/2) * 0.20 kg * (1.5 m/s)^2 = (1/2) * 0.20 kg * v1'^2 + (1/2) * 0.40 kg * v2'^2 + (1/2) * 0.60 kg * v3'^2
0.225 kg m^2/s^2 = 0.1 kg v1'^2 + 0.2 kg v2'^2 + 0.3 kg v3'^2

Now, we have a system of simultaneous equations that we can solve to find v1' and v2'. Since v3' is not mentioned in the problem statement, we won't be able to determine its value.

This is how you can find the final speeds of cart 1 and cart 2 in this scenario.

To find the final speeds of the carts, we can use the principle of conservation of linear momentum. According to this principle, the total momentum before the collision is equal to the total momentum after the collision. Mathematically, it can be written as:

m1*v1i + m2*v2i + m3*v3i = m1*v1f + m2*v2f + m3*v3f (Equation 1)

where m1, m2, and m3 are the masses of carts 1, 2, and 3 respectively, v1i, v2i, and v3i are the initial velocities of the carts, and v1f, v2f, and v3f are the final velocities of the carts.

Given:
m1 = 0.20 kg (mass of cart 1)
m2 = 2m (mass of cart 2)
m3 = 3m (mass of cart 3)
v1i = 1.5 m/s (initial velocity of cart 1)
v2i = 0 m/s (initial velocity of cart 2)
v3i = 0 m/s (initial velocity of cart 3)

To find the final speeds of the carts, we need to solve Equation 1 for v1f, v2f, and v3f.

1. Finding the final speed of cart 1 (v1f):

From Equation 1, we have:

0.20 kg * 1.5 m/s + 2m * 0 m/s + 3m * 0 m/s = 0.20 kg * v1f + 2m * v2f + 3m * v3f

0.30 kg m/s = 0.20 kg * v1f

v1f = 0.30 kg m/s / 0.20 kg

v1f = 1.5 m/s

Therefore, the final speed of cart 1 (v1f) is 1.5 m/s.

2. Finding the final speed of cart 2 (v2f):

From Equation 1, we have:

0.20 kg * 1.5 m/s + 2m * 0 m/s + 3m * 0 m/s = 0.20 kg * 1.5 m/s + 2m * v2f + 3m * v3f

0.30 kg m/s = 0.30 kg m/s + 2m * v2f

v2f = 0 m/s

Therefore, the final speed of cart 2 (v2f) is 0 m/s.

3. Finding the final speed of cart 3 (v3f):

From Equation 1, we have:

0.20 kg * 1.5 m/s + 2m * 0 m/s + 3m * 0 m/s = 0.20 kg * 1.5 m/s + 2m * 0 m/s + 3m * v3f

0.30 kg m/s = 0.30 kg m/s + 3m * v3f

v3f = 0 m/s

Therefore, the final speed of cart 3 (v3f) is 0 m/s.

In summary:
- The final speed of cart 1 (v1f) is 1.5 m/s.
- The final speed of cart 2 (v2f) is 0 m/s.
- The final speed of cart 3 (v3f) is 0 m/s.