Question

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

Answer 1

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

This is proved in attached image .

Answer 2

Hence proven that

Tangential acceleration = component of acceleration along velocity.

Explanation:

v = vxiˆ + vyjˆ  

Acceleration a = dvx / dtiˆ + dvy / dtjˆ

Component of a along v will be,

a.v / |v| = vx dvx  / dt + vy.dvy / dt ÷ √ vx^2 + vy^2

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

or at = dv / dt = d / dt ( √vx^2  + vy^2 )

at = 1 / 2 √vx^2 + vy^2 [ 2vx . dvx / dt + 2vy .dvy / dt]

or at =  vx dvx  / dt + vy.dvy / dt ÷ √ vx^2 + vy^2

Form Equations. (1) and (2) we can see that

Tangential acceleration = component of acceleration along velocity.