Question

Kinematics Differential question: -

Answer 1

A particle's position on the x-axis is given by\[\color{blue}{x=4-27t+t^3.}\]With 'x' in meters & 't' in seconds.
Is there ever a time when\[ \color{red}{v=0 \space ?}\]

Answer 2

I have done as below: -

Answer 3

\[\LARGE{v=\frac{dv}{dt}.}\]Thus; \[v=\frac{d}{dt}(4-27t+t^3.)\]\[\implies v=-27+3t^2.\]So according to question v=0.
Therefore; \[v=-27+3t^2=0\]\[\implies t=\pm 3.\]but time can never be negative so
t=+3 sec. is our answer.

Answer 4

But my book is saying that both values of time are answers. The book mean to say: -\[\color{red}{v=\pm 3. \checkmark}\]
But how can that be actually?

Answer 5

you meant \[v=\frac{\text dx}{\text dt}\]
and \[t=\pm 3\]

Answer 6

It is possible for time to be negative - it just depends on your reference point. Say I consider two points in time, right now (t = 0), and 3 seconds from now (t = 3). Now say I am at that point in time 3 seconds from now, t = 3. I would like to now consider THIS to be my new t = 0, which means the old t = 0 is now t = -3, or 3 seconds ago.

Since the velocity of the particle is modeled by a parabola, there are two points where v = 0. In this case, you just have designated a time point in the middle of the particle's movement to be t = 0 rather than the time point at the very beginning of its trip, meaning you have both "positive" and "negative" time values.

Answer 7

ya @UnkleRhaukus I meant only t=+3 but book meant t=+-3 & @matt101 plz wait a minute while I read ur reply.

Answer 8

so u mean @matt101 that the particle is zero both 3sec. before & 3sec. after a particular time.

Answer 9

?

Answer 10

has zero velocity*

Answer 11

you have done this correctly,
negative values of t reefer to past events,
while events that occur in the future will occur in t seconds

Answer 12

my book has a comment too about this point.

Answer 13

That's right. A parabola extends to infinity in both directions, so every velocity will occur at two points in time (except the max/min velocity). Whether those time points are both positive, both negative, or one of each depends on the horizontal translation of the parabola (i.e. the point in time you designate as t = 0).

Answer 14

Plz read my book's comment about this point.
Then make me understand plz:)

Answer 15

Does it mean that 3sec. before & 3sec. after a particular time, particles velocity is zero?

Answer 16

yes

Answer 17

but what is that particular time?

Answer 18

that particular time is t=0,