Match the following columns. ( Column I Column II (A) (d v)/(dt)...

- Part A: ($\frac{d y}{d t}$) - This is the time derivative of velocity. By definition, the rate of change of velocity is acceleration. Since velocity is a vector, its derivative is also a vector, representing acceleration. - Therefore, (A) matches with (p) Acceleration. - Part B: ($\frac{d|v|}{d t}$) - This is the time derivative of the magnitude of velocity (speed). This quantity is the rate of change of speed. This is NOT the same as the magnitude of acceleration. For example, in uniform circular motion, the speed is constant, so ($\frac{d|v|}{d t}=$0), but the object is accelerating because its direction is changing. The magnitude of acceleration is nonzero in this case. - Therefore, (B) matches with (q) Magnitude of acceleration. - Part C: ($\frac{d r}{d t}$) - This is the time derivative of the position vector. By definition, the rate of change of position is velocity. - Therefore, (C) matches with (r) Velocity. - Part D: ($\frac{|d r|}{d t}$) - The term ($\frac{|d r|}{d t}$) is the magnitude of the rate of change of the position vector. As we know, ($\frac{d r}{d t}$) is the velocity, so taking its magnitude gives us the magnitude of velocity or speed. - Therefore, (D) matches with (s) Magnitude of velocity.