Two masses, 0.600 kg and 0.300 kg, begin uniform motion at...

Two masses, $0.600 \mathrm{~kg}$ and $0.300 \mathrm{~kg}$, begin uniform motion at the same speed, $0.800 \mathrm{~m} / \mathrm{s}$, from the origin at $t=0$ and travel in the directions shown in Figure $\mathrm{P} 9.48 .$ (a) Find the velocity of the center of mass in unitvector notation. (b) Find the magnitude and direction of the velocity of the center of mass. (c) Write the position vector of the center of mass as a function of time.

Two masses, $0.600 \mathrm{~kg}$ and $0.300 \mathrm{~kg}$, begin uniform motion at the same speed, $0.800 \mathrm{~m} / \mathrm{s}$, from the origin at $t=0$ and travel in the directions shown in Figure $\mathrm{P} 9.48 .$
(a) Find the velocity of the center of mass in unitvector notation. (b) Find the magnitude and direction of the velocity of the center of mass. (c) Write the position vector of the center of mass as a function of time.

Key Concepts

Center of Mass

The center of mass of a system is the weighted average of the positions of all the masses in the system, where the weights are the masses. This concept is vital when analyzing the overall motion of a system because it represents the point at which external forces can be considered to act, simplifying the study of dynamics.

Momentum and Center-of-Mass Velocity

The velocity of the center of mass is determined by the total momentum of the system divided by the total mass. This concept links linear momentum and mass distribution, since individual mass momenta (mass times velocity vectors) combine to yield a single effective 'average' velocity for the entire system.

Vector Addition and Unit Vectors

When dealing with motion in more than one dimension, vector addition using unit vectors (such as i-hat and j-hat) becomes essential. Representing velocities and positions as vectors helps in accurately defining their magnitudes and directions, and simplifies the process of combining different components of motion.

Uniform Linear Motion

Uniform linear motion refers to motion where an object travels at a constant velocity, meaning its speed and direction remain unchanged. This concept allows for the straightforward calculation of position as a function of time by integrating the constant velocity, leading to a linear equation of motion.

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