Chapter 07, Problem 48
Two particles are moving along the x axis. Particle 1 has a mass m1 and a velocity v1 = +5.0 m/s. Particle 2 has a mass m2 and a velocity v2 = -7.9 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.
We can find the ratio m1/m2 of the masses of the particles using the formula for the velocity of the center of mass:
V_cm = (m1 * v1 + m2 * v2) / (m1 + m2)
Since the velocity of the center of mass is zero, we can set up the equation:
0 = (m1 * 5.0) + (m2 * -7.9) / (m1 + m2)
0 = (5 * m1 - 7.9 * m2) / (m1 + m2)
Now, we want to find the ratio m1/m2, so we'll manipulate the equation accordingly:
0 = 5 * m1 - 7.9 * m2
5 * m1 = 7.9 * m2
Now, we can find the ratio m1/m2:
m1/m2 = 7.9/5
m1/m2 = 1.58
Therefore, the ratio m1/m2 of the masses of the particles is 1.58.
To find the ratio m1/m2 of the masses of the particles, we can use the concept of the center of mass.
The velocity of the center of mass (Vcm) is given by the equation:
Vcm = (m1 * v1 + m2 * v2) / (m1 + m2)
Since the velocity of the center of mass is zero, we can set this equation equal to zero:
0 = (m1 * v1 + m2 * v2) / (m1 + m2)
Now we can substitute the given values:
0 = (m1 * 5.0 m/s + m2 * -7.9 m/s) / (m1 + m2)
Multiplying both sides of the equation by (m1 + m2) to eliminate the denominator:
0 = m1 * 5.0 m/s + m2 * -7.9 m/s
Rearranging the terms:
7.9 m/s * m2 = 5.0 m/s * m1
Dividing both sides of the equation by 5.0 m/s * m2:
7.9 m/s / 5.0 m/s = m1 / m2
Simplifying:
1.58 = m1 / m2
Therefore, the ratio m1/m2 of the masses of the particles is 1.58.
To find the ratio m1/m2 of the masses of the particles, we can use the concept of calculating the center of mass.
The center of mass (x_cm) of a system of particles is given by the formula:
x_cm = (m1 * x1 + m2 * x2) / (m1 + m2)
In this problem, the center of mass is stationary, which means the velocity of the center of mass (v_cm) is zero. Given the velocities of the particles (v1 and v2), we can relate the velocities to the position of the center of mass.
v_cm = (m1 * v1 + m2 * v2) / (m1 + m2)
Since v_cm = 0, we have:
0 = (m1 * v1 + m2 * v2) / (m1 + m2)
Now we can solve for m1/m2.
Rearranging the equation, we get:
0 = m1 * v1 + m2 * v2
Multiply throughout by (m1 + m2), we have:
0 = m1 * v1 * (m1 + m2) + m2 * v2 * (m1 + m2)
Distribute and rearrange, we get:
0 = m1 * (m1 + m2) * v1 + m2 * (m1 + m2) * v2
Now, substitute the given values:
0 = m1 * (m1 + m2) * 5 + m2 * (m1 + m2) * (-7.9)
Simplify the equation:
0 = 5m1^2 + 5m1m2 - 7.9m1m2 - 7.9m2^2
Combine like terms:
0 = 5m1^2 - 2.9m1m2 - 7.9m2^2
Factor the equation:
0 = (5m1 + 1.3m2)(m1 - 7.9m2)
Since we know mass cannot be negative, we can ignore the second factor, m1 - 7.9m2.
Therefore, solving for the first factor:
5m1 + 1.3m2 = 0
Now, we have a single equation with two variables. We need additional information to solve for m1/m2.