Two particles of equal mass have velocities v⃗1=2 î m s^-1 and v̅2=2 ĵ m s^-1, First particl

step 1 In this question, we have given two particles of equal masses and we have given the velocities o

Two particles of equal mass have velocities v⃗1=2 î m s^-1...

Two particles of equal mass have velocities $\vec{v}_{1}=2 \hat{i} \mathrm{~m} \mathrm{~s}^{-1}$ and $\bar{v}_{2}=2 \hat{j} \mathrm{~m} \mathrm{~s}^{-1}$, First particle has an acceleration $\vec{a}_{1}=(3 \hat{i}+3 \hat{j}) \mathrm{m} \mathrm{s}^{-2}$ while the acceleration of the other particle is zero. The centre of mass of the two particles moves in a path of (a) straight line (b) parabola (c) circle (d) ellipse

Two particles of equal mass have velocities $\vec{v}_{1}=2 \hat{i} \mathrm{~m} \mathrm{~s}^{-1}$ and $\bar{v}_{2}=2 \hat{j} \mathrm{~m} \mathrm{~s}^{-1}$, First particle
has an acceleration $\vec{a}_{1}=(3 \hat{i}+3 \hat{j}) \mathrm{m} \mathrm{s}^{-2}$ while the acceleration of the other particle is zero. The centre of mass of the two particles moves in a path of
(a) straight line
(b) parabola
(c) circle
(d) ellipse

Key Concepts

Centre of Mass Motion

The centre of mass of a system of particles is the weighted average position of the particles, where the weights are typically their masses. In dynamics, it is useful because the motion of the center of mass provides insight into the overall motion of the system, regardless of the internal forces acting between the particles.

Constant Acceleration in Kinematics

Motion under constant acceleration is governed by kinematic equations that describe how positions and velocities change over time. When a body experiences a constant acceleration, its velocity changes linearly with time and its displacement includes a quadratic term in time, forming trajectories such as parabolic paths unless the velocity and acceleration are collinear.

Vector Addition in Kinematics

In problems involving velocity and acceleration, vector addition is essential for determining resultant values. The overall velocity or acceleration of a system like the center of mass is determined by summing the individual vectors, which may involve both magnitude and direction. This technique simplifies multi-particle systems into a single effective motion.

Role of Initial Conditions in Determining Trajectories

Initial conditions, such as starting velocity and position, critically influence the subsequent motion of an object under acceleration. When the initial velocity and the constant acceleration are aligned, the resulting trajectory is a straight line. If they are not collinear, the trajectory becomes curved, highlighting the importance of analyzing the directional relationship between initial conditions and applied forces.

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