A particle moves in a straight line with a velocity given by $v = \alpha t + \beta t^3$, where $\alpha$ and $\beta$ are constants. If its acceleration at $t=1$ second is twice its acceleration at $t=0$, what is the relation between $\alpha$ and $\beta$?

The minimum phase difference between two simple harmonic oscillations, ${y_1} = \frac{1}{2}\sin \,\,\omega t + \frac{{\sqrt 3 }}{2}\cos \,\,\omega t$ ${y_2} = \sin \,\,\omega t + \cos \,\,\omega t,$ is

The amplitude of SHM $\;y = 2\left( {\sin 5\,\,{\rm{\pi }}t + \sqrt 2 {\rm{cos}}\,\,{\rm{\pi }}t} \right)$ is

A ball is thrown vertically upwards with an initial velocity of 10 m/s. What is its velocity at the highest point of its trajectory?

A particle executing SHM has a maximum speed of $31.4$ cm/s and a maximum acceleration of $5$ m/s$^2$. Find the amplitude of oscillation.

A particle is projected vertically upwards with an initial speed 'u'. At a height 'h' above the ground, its speed is 'v'. What is the maximum height reached by the particle?

The phase difference between displacement and acceleration of a particle in a simple harmonic motion is:

A particle executes SHM in a line 4 cm long. Its velocity when passing through the centre of line is $12cm/s$ . The period will be

Two particles are executing simple harmonic motion of the same amplitude A and frequency ω along the x-axis. Their mean position is separated by distance ${x_0}$ $({x_0} > A)$. If the maximum separation between them is $\left( {{x_0} + A} \right)$, the phase difference between their motions is

The acceleration of a particle in SHM is $0.8\,\,m{s^{ - 2}}$ when its displacement is 0.2 m. The frequency of oscillation is

A particle's position is given by $\vec r = A\cos \left( {\omega t + \phi } \right)\hat x + A\sin \left( {\omega t + \phi } \right)\hat y$, where $A$, $\omega$, and $\phi$ are constants. What type of motion does the particle exhibit?