Question

Particle Motion

Answer 1

Suppose the velocity of a particle is given by the function
\[v(t)=(t+2)(t+4)^2\]

Answer 2

for \[t \ge0\] where t is measured in minutes and v(t) is measured in inches per minute. Answer the questions that follow.

Answer 3

Find the values of v(3) and v ' (3). Based on the values, describe the speed of the particle at t=3.

Answer 4

On what interval(s) is the particle moving to the left?Right? Show your analysis and justify your answer.

Answer 5

v(3)=245 in/min
v ' (3)=119 in/min

Answer 6

I am stuck on the description of the speed of the particle at t = 3
I never took physics so particle motion is not something I am good at

Answer 7

v = negative means the particle is moving in opposite direction (left)

Answer 8

Well your figure is perfectly correct. :) (As expected)

Answer 9

(-2, infinity) particle is moving right
(- infinity, -4) (-4, -2) particle is moving left
particle stop at t=-4 and t=-2

Answer 10

not sure about those conclusions

Answer 11

Correct !

Answer 12

Perfectly correct @precal :)

Answer 13

Great work

Answer 14

but I am still stuck on the description of the speed at t=3
v(3)=245 in/min is that the velocity? 245 inches per min at 3 minutes

Answer 15

why ask about v ' (3)? I know this is acceleration

Answer 16

or should I say for v(3)
speed is increasing 245 inches per min ?

Answer 17

v(3) = 245 tells you the distance is changing at 245 inches per min
v'(3) = positive tells you the velocity is increasing

Answer 18

Overall :
at t = 3, the velocity is 245 inches per min, and it is increasing at a rate of 119 inches/min/min

Answer 19

is v ' (t) inches/min^2? I struggle with units as well. I don't recall seeing inches/min/min

Answer 20

The change of position is velocity
The change of velocity is acceleration

So if you're driving your car and look down at your speed after a few minutes of driving and it says 30mph then this means v( a few minutes) = 30 mph.

Acceleration is when you speed up from 30 to 40 or break to slow down from 30 to 20 mph. You feel acceleration when you're in your car speeding up pushing you against the back of your seat or jerking your head forward when you stop suddenly, but you don't feel velocity. This is the whole "object in motion tends to stay in motion". After all, the earth is spinning and revolving around the sun at the same time and yet we don't feel anything! That's because it's not accelerating.

Answer 21

change in position : inches / min
change in velocity : (inches/min) / min

Answer 22

thanks I appreciate all of your help and input

Answer 23

To state in calculus terms: change of position is velocity:\[\Large \frac{dx}{dt}=v\]Change of velocity is acceleration:\[\Large \frac{dv}{dt}=a\]

So plug in the v from the first into the derivative of the second:\[\Large \frac{d}{dt} \left( \frac{dx}{dt} \right) = \frac{d^2x}{dt^2}=a\]
Notice that the units match up in all cases here. Every time we take the derivative with respect to time, it's dividing by time which matches our units and looks like we're adding another (dt) on bottom.

Answer 24

^^
I find these short videos entertaining
https://www.youtube.com/watch?v=I3GlErUi2q0#t=61

Answer 25

guess they are not working at the moment. @ganeshie8