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Instantaneous Velocity and Acceleration

Let's discuss a problem useful from Physics Olympiad based on Instantaneous Velocity and Acceleration.

The Problem:

Let (\vec{v}) and (\vec{a}) be instantaneous velocity and the acceleration respectively of a particle moving in a plane. The rate of change of speed (dv/dt) of the particle is:
(a) (|a|)
(b) ((v.a)/|v|)
(c) the component of (\vec{a}) in the direction of (\vec{v})
(d) the component of (\vec{a}) perpendicular to (\vec{v})

Solution:

Let us consider (v^2=v_x^2+v_y^2).
We differentiate the above equation.
(\frac{dv}{dt})=((v_xa_x+v_ya_y)v)=(\frac{v.a}{v}).
Hence, the correct option will be B along with C since the component of a is in the direction of v.

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