Two particles are moving along the x axis. Particle 1 has a mass m1 and a velocity v1 = +4.3 m/s. Particle 2 has a mass m2 and a velocity v2 = -6.3 m/s. The velocity of the center of mass of these two particles is zero. In other words, the center of mass of the particles remains stationary, even though each particle is moving. Find the ratio m1/m2 of the masses of the particles.

Question

Two particles are moving along the x axis. Particle 1 has a mass m1 and a velocity v1 = +4.3 m/s. Particle 2 has a mass m2 and a velocity v2 = -6.3 m/s. The velocity of the center of mass of these two particles is zero. In other words, the center of mass of the particles remains stationary, even though each particle is moving. Find the ratio m1/m2 of the masses of the particles.

Answer 1

m1/m2 = 4.3/-6.3 = 0.68

Answer 2

To find the ratio of the masses m1/m2, we can use the formula for the center of mass velocity of a system of particles:

v_cm = (m1*v1 + m2*v2) / (m1 + m2)

Since the center of mass velocity is zero:

0 = (m1*v1 + m2*v2) / (m1 + m2)

Multiplying both sides by (m1 + m2) gives:

0 = m1v1 + m2v2

We are given that v1 = +4.3 m/s and v2 = -6.3 m/s, so we can substitute these values in the equation:

0 = m1*(+4.3 m/s) + m2*(-6.3 m/s)

Simplifying the right side:

0 = 4.3m1 - 6.3m2

Rearranging the equation:

6.3m2 = 4.3m1

Dividing both sides by 4.3:

m2/m1 = 4.3/6.3

So the ratio of the masses m1/m2 is approximately 0.683.

Answer 3

To solve this problem, we need to use the concept of the center of mass. The center of mass is the average position of all the particles in a system, weighted by their masses. In this case, we are given that the velocity of the center of mass is zero, which means that the net momentum of the system is zero.

The momentum of a particle is given by the product of its mass and velocity: p = m * v. The total momentum of the system is the sum of the momenta of each particle, so we have:

p1 + p2 = 0

Since the velocities are given, we can substitute the expressions for momentum:

m1 * v1 + m2 * v2 = 0

Let's substitute the given values into this equation:

m1 * 4.3 m/s + m2 * (-6.3 m/s) = 0

Now, we can solve this equation for the ratio m1/m2.

Divide both sides of the equation by m2:

m1 * (4.3 m/s) + (-6.3 m/s) = 0

Now, let's isolate m1:

m1 * (4.3 m/s) = 6.3 m/s

Divide both sides by 4.3 m/s to solve for m1:

m1 = (6.3 m/s) / (4.3 m/s)

Simplify the right side:

m1 ≈ 1.465

Therefore, the ratio m1/m2 of the masses of the particles is approximately 1.465.