Question

The position of a particle as it moves along the x axis is given for t > 0 by x = (t^3-3t^2+6t)m. Where is the particle when it achieves its minimum speed (after t = 0)?

Answer 1

The particle achieves its minimum speed when the acceleration is 0.

The objective is to get from x(t) to a(t), this involves derivation.

x(t) -- once derived --> v(t) -- once derived--> a(t).

You just need to take 2 derivatives of the equation x(t)=t³-3t²+6t. Take the first drivative with respect to t to get v(t)=3t²-6t+6. Again take another derivative with respect to t to get a(t)= 6t-6.

Now set this equal to 0 to find the time. a(t)=6t-6=0 solving for t=6/6= 1 second

Now go back to x(t) and substitute t=1 to find distance travelled. x(1)=1³-3*1²+6*1 = 1-3+6= 4 m. You can even find the speed at that moment by just substituting v(1)=3-6+6=3 m/s.

The minimum speed for you particle is 3 m/s when t=1s and x=4m.

Hopes this helps