Question

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

Answer 1

Answer:

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.

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,

$v(t)=\frac{d}{dt}(s(t))$

Substitute for s(t), we get

$v(t)=\frac{d}{dt}(4t^3+36t^2+96t)$

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,

$a(t)=\frac{d}{dt}(v(t))$

Substitute v(t), we get,

$a(t)=\frac{d}{dt}(12t^2+72t+96)$

we get,

$a(t)=24t+72$

The acceleration of a particle is,

$a(t)=24t+72$