JEE mains physics class 11th

Question

In a two dimensional motion of a body prove that tangential acceleration is nothing but component of acceleration along velocity.

Answer 1

Let velocity of the particle be,

\(\vec v = v_x\hat i + v_y\hat j\)

Acceleration \(\vec a = \frac{dv_x}{dt} \hat i + \frac{dv_y}{dt}\hat j\)

Component of \(\vec a\) along \(\vec v\) will be,

\(\frac{\vec a.\vec v}{|\vec v|} =v_x. \frac{dv_x}{dt} + v_y. \frac{dv_y}{dt} \quad....(i)\)

Further, tangential acceleration of particle is rate of change of speed.

\(a_t = \frac{dv}{dt} = \frac d{dt} \left(\sqrt {v_x^2 + v_y^2}\right)\)

\(a_t = \frac 1{2\sqrt{v_x^2 + v_y^2}} \left[ 2v_x. \frac{dv_x}{dt} + 2v_y \frac{db_y}{dt}\right]\)

\(a_t = \frac{v_x. \frac{dv_x}{dt} + v_y.\frac{dv_y}{dt}}{\sqrt{v_x^2 + v_y^2}}\)

From Eqs. (i) and (ii),

\(a_t = \frac{\vec a .\vec v}{|\vec v|}\)

Tangential acceleration = component of acceleration along velocity.