(a) A particle is moving eastward with a velocity of 5 m/s. If in 10 s the velocity changes by `5 m//s` northwards. What is the average acceleration in this time ?
(b) What is the retardation of a moving particle if the relation between time and position is, `t = Ax^(2) + Bx`? (where A and B are appropriate constants)

Explanation

`Delta vec(v)_(2) - vec(v)_(1) = vec(v)_(2) + (-vec(v)_(1))`
i.e., `Delta vec(v) = (sqrt(5^(2) + 5^(2))) N - W`
i.e., `Delta vec(v) = 5 sqrt2 N - W`
So, `vec(a)_(av) = (Delta vec(v))/(Delta t) = (5sqrt2)/(10) N - W`
i.e., `vec(a)_(av) = (1)/(sqrt2) (m//s^(2)) N - W`

(b) As `t = Ax^(2) + Bx`
So, `(dt)/(dx) = 2Ax + B` or `v = (2Ax + B)^(-1)`...(i)
Now by chain rule,
`a = (dv)/(dt) = (dv)/(dx).(dx)/(dt) = v (dv)/(dx)` [as `(dx)/(dt) = v`]
So, `a = v (d)/(dx) (2Ax + B)^(-1)` [from Eqn. (i)]
or `a = v (-1) (2Ax + B)^(-2) (2A)`
or `a = - 2A (2Ax + B)^(-3)` [as `v = (2Ax + B)^(-1)`]
So, retardation `= -a 2A (2Ax + B)^(-3)`