The displacement of a particle from a fixed source is given as x(t) = t³ − 2t² − 4t + 5 (m). How far is the…

Question

The displacement of a particle from a fixed source is given as x(t) = t³ − 2t² − 4t + 5 (m). How far is the particle from the source when it goes to momentary rest?

ScanSolve AI Tutor · Accepted Answer

Here are the solutions to problems 9 and 10:

9. The displacement of a particle from a fixed source is given as $x(t) = t^3 - 2t^2 - 4t + 5 \text{ (m)}$ . How far is the particle from the source when it goes to momentary rest?

To find when the particle is at momentary rest, we first need to find its velocity by differentiating the displacement function with respect to time. Momentary rest occurs when the velocity is zero.

Find the velocity function $v(t)$ .
$v(t) = \frac{dx}{dt} = \frac{d}{dt}(t^3 - 2t^2 - 4t + 5)$ $v(t) = 3t^2 - 4t - 4 \text{ m/s}$
Set $v(t) = 0$ to find the time(s) when the particle is at rest.
$3t^2 - 4t - 4 = 0$
Using the quadratic formula $t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ :
$t = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(3)(-4)}}{2(3)}$ $t = \frac{4 \pm \sqrt{16 + 48}}{6}$ $t = \frac{4 \pm \sqrt{64}}{6}$ $t = \frac{4 \pm 8}{6}$
Two possible values for $t$ :
$t_1 = \frac{4 + 8}{6} = \frac{12}{6} = 2 \text{ s}$ $t_2 = \frac{4 - 8}{6} = \frac{-4}{6} = -\frac{2}{3} \text{ s}$
Since time cannot be negative, we consider $t = 2

The displacement of a particle from a fixed source is given as x(t) = t³ − 2t² − 4t + 5 (m). How far is the particle from the source when it goes to momentary rest?

Step-by-step solution

Find the velocity function $v(t)$ .

Set $v(t) = 0$ to find the time(s) when the particle is at rest.

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More Physics Questions

The displacement of a particle from a fixed source is given as x(t) = t³ − 2t² − 4t + 5 (m). How far is the particle from the source when it goes to momentary rest?

Step-by-step solution

Find the velocity function v(t)v(t)v(t).

Set v(t)=0v(t) = 0v(t)=0 to find the time(s) when the particle is at rest.

Still stuck on this one?

More Physics Questions

Find the velocity function $v(t)$ .

Set $v(t) = 0$ to find the time(s) when the particle is at rest.