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
Jun 13, 2017

Average velocity: 6.01 xx 10^3 "m/s"

Velocity at time t = 0 "s": 0 "m/s"

Velocity at t = 10 "s": 2.40 xx 10^4 "m/s"

Explanation:

I'll assume you mean the average velocity from t = 0 to t = 10 "s".

We're given the components of the particle's acceleration, and asked to find the average velocity over the first 10 seconds of its motion:

vecv_"av" = (Deltavecr)/(10"s")

where

  • v_"av" is the magnitude of the average velocity, and

  • Deltar is the change in postion of the object (from 0 "s" to 10 "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:

vecv = (t^3)hati + (5/2t^2)hatj - (2t^4 + 400t)hatk (velocity)

Second integration:

vecr = (1/4t^4)hati + (5/6t^3)hatj - (2/5t^5 + 200t^2)hatk (position)

The initial position is assumed to be at the origin, so let's plug in 10 for t in the position equation:

vecr = (2500)hati + (2500/3)hatk - (60000)hatk

We can then split the average velocity equation into components:

v_"av-x" = (Deltax)/(10"s") = (2500"m")/(10"s") = color(red)(250 color(red)("m/s"

v_"av-y" = (Deltay)/(10"s") = (2500/3"m")/(10"s") = color(blue)(250/3 color(blue)("m/s"

v_"av-z" = (Deltaz)/(10"s") = (-60000"m")/(10"s") = color(green)(-6000 color(green)("m/s"

Using these components, we can find the magnitude of the average velocity vector:

v_"av" = sqrt((v_"av-x")^2 + (v_"av-y")^2 + (v_"av-z")^2)

= sqrt((250"m/s")^2 + (250/3"m/s")^2 + (-6000"m/s")^2)

= color(purple)(6.01 xx 10^3 color(purple)("m/s"

(Here's the instantaneous velocity section) .

To find the instantaneous velocities at t = 0 and t = 10 "s", let's first plug in these times into the previously integrated velocity equation:

  • 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"