Question

. The displacement (x) of a particle depends on time t
as x=αt² - βt³. Choose the incorrect statements
from the following.
(a) The particle never returns to its starting point
(b) The particle comes to rest after time 2α/3β
(c) The initial velocity of the particle is zero
(d) The initial acceleration of the particle is zero ​

Answer 1

Answer:

Particle will return to its starting point when,x=αt

2

−βt

3

=0

or,

t=

β

α

Velocity=v=

dt

dx

=2tα−3βt

2

.....(1)

at t=0,v=0so the initial velocity zero.

Acceleration=a=

dt

dv

=2α−6tβ.....(2)

at t=0,a=2α so initial acceleration does not zero.

The particle will come to rest when, 2tα−3βt

2

=0 from (1)

or,t=

3β

2α

At,t=

3β

α

,a=2α−6β(

3β

α

)=0

so net force=ma=0 ,thus no net force act on the particle when t=

3β

α

Answer 2

Answer:

both a and d are incorrect

Explanation:

for returning to the starting point,

displacement x should be zero

for x=0,

αt² - βt³=0

so t= 0 or α/β

so after t = α/β the particle will return to its starting point.

and initial acceleration of the particle is 2α

(by double derivative of x)