What is the velocity of a particle for t=0 to t=10 whos acceleration is veca =3t^2 hati+5t hatj-(8t^3+400)hatk?
1 Answer
Average velocity:
Velocity at time
Velocity at
Explanation:
I'll assume you mean the average velocity from
We're given the components of the particle's acceleration, and asked to find the average velocity over the first
where
-
v_"av" is the magnitude of the average velocity, and -
Deltar is the change in postion of the object (from0 "s" to10 "s" ).
We must therefore find the position of the object at these two times.
We have to derive a position equation from this acceleration equation, by integrating it two times:
First Integration:
Second integration:
The initial position is assumed to be at the origin, so let's plug in
We can then split the average velocity equation into components:
Using these components, we can find the magnitude of the average velocity vector:
(Here's the instantaneous velocity section) .
To find the instantaneous velocities at
t = 0 "s"
vecv = ((0"s")^3)hati + (5/2(0"s")^2)hatj - (2(0"s")^4 + 400(0"s"))hatk
= color(red)(0 color(red)("m/s"
t = 10 "s"
vecv = ((10"s")^3)hati + (5/2(10"s")^2)hatj - (2(10"s")^4 + 400(10"s"))hatk
= (1000"m/s")hati + (250"m/s")hatj - (24000"m/s")hatk The magnitude of this velocity is thus
v(10"s") = sqrt((1000"m/s")^2 + (250"m/s")^2 + (-24000"m/s")^2)
= color(blue)(2.40 xx 10^4 color(blue)("m/s"