Given a particle that has the acceleration a(t) = 4e3 meters, a. find the displacement at 2 seconds, assuming an initial position of o & from rest. b. find velocity at 1 second assuming starting from rest.

Hi!

The acceleration of our particle is then a = 4* meters per second square. Also we know that we start at X₀ = O m (i am not sure if you are using a a 0 or a O, but you will see that in this problem it doesn't matter, se i will use a constant O, that can be  ani real number as the initial position) and V₀ = 0 m/s.

Ok, we have the acceleration, so if we want the velocity, we must integrate the acceleration over time.

V(t) = ∫ 4*dt = 4**t + V₀

where we added the constant of integration, who is the inicial velocity, that we already know that is zero.

and for the position, we must integrate again.

X(t) =  ∫ 4**tdt =  4**/2 + X₀

ok, in the part a) we want to know the displacement at 2 seconds.

that is X(2s) - X(0s) = (2000*4 + O) - O meters = 8000 meters.

in the b part we want to know the velocity at 1 second; that is:

V(1s) = 4000*1 m/s = 4000 meters per second.

resonant frequency

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