Object A is moving due east, while object B is moving due north. They collide and stick together in a completely inelastic collision. Momentum is conserved. Object A has a mass of mA = 17.0 kg and an initial velocity of VoA = 8.15 m/s, due east. Object B, however, has a mass of mB = 28.0 kg and an initial velocity of VoB = 4.85 m/s, due north. Find the magnitude of the final velocity of the two-object system after the collision.

Well, it seems like A and B had a little rendezvous and decided to stick together. Let's see what happens.

To find the final velocity of the two-object system, we need to use the law of conservation of momentum. The total initial momentum of the system is equal to the total final momentum of the system.

Now, the momentum of an object is mass times velocity. So, the initial momentum of object A is 17.0 kg * 8.15 m/s (due east), while the initial momentum of object B is 28.0 kg * 4.85 m/s (due north).

To add up the momenta, we have to consider the directions. Since object A is moving east and object B is moving north, we need to use vector addition to determine the direction of the total momentum.

To do this, we can find the components of the initial velocities of A and B using trigonometry. The component of A's initial velocity in the north direction is A_north = 8.15 m/s * sin(90°), while the component of B's initial velocity in the east direction is B_east = 4.85 m/s * cos(90°).

Adding up these components, we get the total initial momentum in the east direction (Px_initial) and the total initial momentum in the north direction (Py_initial).

Now, because momentum is conserved, the total initial momentum in the east direction should be equal to the total final momentum in the east direction (Px_final), and the total initial momentum in the north direction should be equal to the total final momentum in the north direction (Py_final).

Finally, we can find the magnitude of the final velocity of the two-object system by using the Pythagorean theorem.

Now, let me grab my calculator and do the math for you.

To find the magnitude of the final velocity of the two-object system after the collision, we can use the principle of conservation of momentum.

1. First, let's calculate the initial momentum of object A and object B separately.

The initial momentum of object A (pA) is given by the product of its mass (mA) and initial velocity (VoA):
pA = mA * VoA

Substituting the given values:
pA = 17.0 kg * 8.15 m/s

The initial momentum of object B (pB) is given by the product of its mass (mB) and initial velocity (VoB):
pB = mB * VoB

Substituting the given values:
pB = 28.0 kg * 4.85 m/s

2. Since momentum is conserved in the collision, the total momentum before the collision (pTotal_initial) should be equal to the total momentum after the collision (pTotal_final).

pTotal_initial = pA + pB

3. After the collision, the two objects stick together and move as one. Let's denote the final velocity of the combined objects as Vf.

So, the final momentum of the two-object system (pTotal_final) is given by the product of the combined mass (mTotal = mA + mB) and final velocity (Vf):
pTotal_final = mTotal * Vf

4. Now, we can equate the total momentum before and after the collision and solve for Vf.

pTotal_initial = pTotal_final
pA + pB = mTotal * Vf

Substituting the values:
17.0 kg * 8.15 m/s + 28.0 kg * 4.85 m/s = (17.0 kg + 28.0 kg) * Vf

68.55 kg · m/s + 138.8 kg · m/s = 45.0 kg * Vf

207.35 kg · m/s = 45.0 kg * Vf

5. Finally, let's solve for Vf by dividing both sides of the equation by the combined mass of the objects.

Vf = (207.35 kg · m/s) / (45.0 kg)

Vf ≈ 4.61 m/s

Therefore, the magnitude of the final velocity of the two-object system after the collision is approximately 4.61 m/s.

To find the final velocity of the two-object system after the collision, we can use the principle of conservation of momentum.

Momentum is defined as the product of an object's mass and velocity. In this case, since the collision is completely inelastic and the objects stick together, the total momentum before the collision must equal the total momentum after the collision.

The momentum before the collision is given by:

P_initial = mA * VoA + mB * VoB

where mA and mB are the masses of object A and B respectively, and VoA and VoB are their initial velocities.

Substituting the given values:

P_initial = 17.0 kg * 8.15 m/s (east) + 28.0 kg * 4.85 m/s (north)

Now, let's resolve the vectors into their respective x and y components.

The x-component of object A's velocity (Va_x) is its initial velocity (VoA) since it is moving due east. The y-component of object A's velocity (Va_y) is 0, since it is not moving in the y-direction.

The x-component of object B's velocity (Vb_x) is 0, since it is not moving in the x-direction. The y-component of object B's velocity (Vb_y) is its initial velocity (VoB) since it is moving due north.

Now, let's calculate the x and y components of the initial momentum:

Px_initial = mA * Va_x + mB * Vb_x = 17.0 kg * 8.15 m/s (east) + 28.0 kg * 0 = 138.55 kg * m/s (east)

Py_initial = mA * Va_y + mB * Vb_y = 17.0 kg * 0 + 28.0 kg * 4.85 m/s (north) = 135.8 kg * m/s (north)

The total momentum before the collision is the vector sum of its x and y components:

P_initial = sqrt(Px_initial^2 + Py_initial^2)

P_initial = sqrt((138.55 kg * m/s)^2 + (135.8 kg * m/s)^2)

P_initial ≈ 196.35 kg * m/s

Since momentum is conserved, the total momentum after the collision must be equal to the initial momentum.

P_final = P_initial

Now, let's calculate the magnitude of the final velocity using the principle of conservation of momentum:

P_final = m_total * V_final

where m_total is the combined mass of the two objects and V_final is their final velocity.

Since the objects stick together after the collision, the combined mass is:

m_total = mA + mB

Substituting the given values:

m_total = 17.0 kg + 28.0 kg = 45.0 kg

Now we can solve for the final velocity:

V_final = P_final / m_total = (196.35 kg * m/s) / 45.0 kg

V_final ≈ 4.36 m/s

Therefore, the magnitude of the final velocity of the two-object system after the collision is approximately 4.36 m/s.