Question

please help me work through this! I'm just a bit stuck and think help would make me better understand

Answer 1

Given equation of motion of a particle is,

`s(t)=4t^3+36t^2+96t`

To find the velocity function of t, value of t when the velocity is zero and acceleration of the particle.

Step-by-step explanation:

we know that, velocity of a particle is rate of change of the object’s position with respect to a frame of reference and time.

we get,

`v(t)=(d)/(dt)(s(t))`

Substitute for s(t), we get

`v(t)=(d)/(dt)(4t^3+36t^2+96t)`

Differenting we get,

`v(t)=12t^2+72t+96`

Required velocity function is,

`v(t)=12t^2+72t+96`

when velocity equal to 0, we get

`12t^2+72t+96=0`

Dividing by 12, we get

`t^2+6t+8=0`

we get,

`t^2+4t+2t+8=0`
`t(t+4)+2(t+4)=0`

Taking (t+4) as common we get,

`(t+4)(t+2)=0`

we get,

`t=-4,-2`

The values of t are,

t=-4 and t=-2

To find acceleration of a particle.

we know that,

Acceleration a(t) is the rate at which velocity (speed) is changing.

we get,

`a(t)=(d)/(dt)(v(t))`

Substitute v(t), we get,

`a(t)=(d)/(dt)(12t^2+72t+96)`

we get,

`a(t)=24t+72`

The acceleration of a particle is,

`a(t)=24t+72`