Question

A particle is at rest at t = 0. If acceleration of the
particle is given as a = sinnt + cositt, in Sl units,
then the maximum speed of particle is​

Answer 1

A particle is at rest at t = 0,

means, initial velocity of particle , u = 0

now acceleration of the particle is given as, $\bf{a=sin\pi t+cos\pi t}$

we know, acceleration is the rate of change of velocity with respect to time.

so, a = dv/dt = sinπt + cosπt

or, $\int\limits^v_0{dv}=\int\limits^t_0{(sin\pi t+cos\pi t)}\,dt$

or, v = -1/π(cosπt + sinπt)

= sinπt - cosπt

= √2/π [ 1/√2 × sinπt - 1/√2 × cosπt ]

= √2/π [ cosπ/4 × sinπt - sinπ/4 × cosπt ]

we know formula, sinA.cosB - cosA.sinB = sin(A - B)

so, cosπ/4 × sinπt - sinπ/4 × cosπt = sin(πt - π/4)

then, v = √2/πsin(πt - π/4)

hence, maximum value of v will be √2/π m/s.

hence, maximum speed of particle is √2/π m/s.