Question

Two objects with masses \(m_{1}\) and \(m_{2}\) are moving along the \(x\) -axis in the positive direction with speeds \(v_{1}\) and \(v_{2}\), respectively, where \(v_{1}\) is less than \(v_{2}\). The speed of the center of mass of this system of two bodies is a) less than \(v_{1}\). b) equal to \(v_{1}\). c) equal to the average of \(v_{1}\) and \(v_{2}\). d) greater than \(v_{1}\) and less than \(v_{2}\). e) greater than \(v_{2}\).

Short answer

Question: The speed of the center of mass of a system of two objects with masses \(m_{1}\) and \(m_{2}\) moving along the x-axis in the positive direction with speeds \(v_{1}\) and \(v_{2}\) (with \(v_{1} < v_{2}\)) is: a) Equal to \(v_{1}\) b) Equal to \(v_{2}\) c) Less than \(v_{1}\) and greater than \(v_{2}\) d) Greater than \(v_{1}\) and less than \(v_{2}\) Answer: d) Greater than \(v_{1}\) and less than \(v_{2}\).

Step-by-step solution

Recall the formula for center of mass

The center of mass of a system of particles can be found using the formula: x_cm = (m_1 * x_1 + m_2 * x_2) / (m_1 + m_2) Additionally, the velocity of the center of mass can be found using the same weights for the velocities, like so: v_cm = (m_1 * v_1 + m_2 * v_2) / (m_1 + m_2)

Determine the relationship between v1 and v2

To find how v_cm compares to v_1 and v_2, note that v_1 < v_2. This relationship between the velocities will be important when we evaluate the expression for the velocity of the center of mass.

Analyze the velocity of the center of mass

v_cm = (m_1 * v_1 + m_2 * v_2) / (m_1 + m_2) Now, since v_1 and v_2 are both positive, the entire numerator (m_1 * v_1 + m_2 * v_2) must also be positive. In addition, both masses m_1 and m_2 must also be positive, which means the denominator (m_1 + m_2) must also be positive. Since we have a fraction where both the numerator and denominator are positive, we can determine that: v_1 * (m_1 / (m_1 + m_2)) < (m_1 * v_1 + m_2 * v_2) / (m_1 + m_2) < v_2 * (m_2 / (m_1 + m_2)) This inequality implies: v_1 < v_cm < v_2

Conclude and choose the correct answer

Based on our analysis, we determined that the velocity of the center of mass is greater than v_1 and less than v_2. Therefore, the correct answer is: d) greater than \(v_{1}\) and less than \(v_{2}\).