What is the instantaneous velocity of an object moving in accordance to f(t)=(t+2,t+4) at t=1?

Question

Curtiello69r · Accepted Answer

Take the derivative of t+2. You should get:12t+2Plug in 1 to get 122.Now take the derivative of t+4. You should get 1 so plugging in 1 will get you 1.Now that you know that it has a velocity of 122. In the x direction and 1 in the y direction, you need to find the resulting velocity from that. Use Pythagorean theorem:a2+b2=c2(122)2+12=c214⋅2+1=c2c=98

Jeremiah Dickerson · Accepted Answer

Instantaneous velocity is given by the vector:
$\vec{v} = ⟨ \frac{\sqrt{3}}{6}, 1 ⟩$ , or, $\frac{\sqrt{3}}{6} \hat{i} + \hat{j}$ , or, $(\begin{matrix} \frac{\sqrt{3}}{6} \\ 1 \end{matrix})$
Explanation:
We have:
f(t)=(x(t), y(t)) where $x (t) = \sqrt{t + 1}, y (t) = t + 4$
Then:
$\frac{d x}{d t} = \frac{1}{2} {(t + 2)}^{- \frac{1}{2}} = \frac{1}{2 \sqrt{t + 2}}$
$\frac{d y}{d t} = 1$
So, when t=1:
$\frac{d x}{d t} = \frac{1}{2 \sqrt{t + 1}} = \frac{1}{2 \sqrt{3}} = \frac{\sqrt{3}}{6}$
$\frac{d y}{d t} = 1$
And so the instantaneous velocity is given by the vector:
$\vec{v} = ⟨ \frac{\sqrt{3}}{6}, 1 ⟩$ , or, $\frac{\sqrt{3}}{6} \hat{i} + \hat{j}$ , or, $(\begin{matrix} \frac{\sqrt{3}}{6} \\ 1 \end{matrix})$
If we want the instantaneous speed , it is given by:
$v = | | \vec{v} | |$
$= \sqrt{\frac{3}{36} + 1}$
$= \sqrt{\frac{13}{12}}$
$\approx 1.04083$

What is the instantaneous velocity of an object

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