In the laboratory, a particle of mass 3.16 kg moving at 15.6 m/s to the left collides head-on with a particle of mass 2.84 kg moving at 12.2 m/s to the right. Find the velocity of the center of mass of the system of two particles after the collision

To determine the velocity of the center of mass of the system after the collision, we first need to analyze the masses and velocities of the two particles involved. The concept of center of mass is crucial here, as it provides insight into the overall motion of a system of particles. Let's break this down step by step.

Step 1: Define the Masses and Velocities

We have two particles with the following characteristics:

• Mass of particle 1 (m1): 3.16 kg, moving to the left at 15.6 m/s.
• Mass of particle 2 (m2): 2.84 kg, moving to the right at 12.2 m/s.

Step 2: Assign Directions

To simplify calculations, we can assign a direction to our velocities. Let's consider left as negative and right as positive:

• Velocity of particle 1 (v1) = -15.6 m/s
• Velocity of particle 2 (v2) = +12.2 m/s

Step 3: Calculate the Total Momentum

The total momentum of the system before the collision is given by the sum of the momenta of both particles. Momentum (p) is calculated using the formula:

p = mass × velocity

So, the total momentum (P_total) before collision can be calculated as follows:

P_total = (m1 × v1) + (m2 × v2) = (3.16 kg × -15.6 m/s) + (2.84 kg × 12.2 m/s)

Step 4: Compute Each Component

Now, let's calculate each term:

• For particle 1: 3.16 kg × -15.6 m/s = -49.296 kg·m/s
• For particle 2: 2.84 kg × 12.2 m/s = 34.648 kg·m/s

Step 5: Sum the Momentums

Now we can find the total momentum:

P_total = -49.296 kg·m/s + 34.648 kg·m/s = -14.648 kg·m/s

Step 6: Calculate the Total Mass

The total mass of the system (M_total) is simply the sum of the masses of the two particles:

M_total = m1 + m2 = 3.16 kg + 2.84 kg = 6.00 kg

Step 7: Find the Velocity of the Center of Mass

The velocity of the center of mass (V_cm) can be calculated using the total momentum and total mass:

V_cm = P_total / M_total

Substituting in the values:

V_cm = -14.648 kg·m/s / 6.00 kg = -2.44133 m/s

Final Result

The velocity of the center of mass of the system after the collision is approximately -2.44 m/s. This negative sign indicates that the center of mass is moving to the left, which is consistent with the motion of the first particle having a greater momentum magnitude than that of the second particle.

Understanding this process not only helps in solving similar problems but also reinforces the fundamental concepts of momentum and center of mass in a system of particles. If you have any more questions about this topic or related concepts, feel free to ask!