Question

a point traversed half a circle of radius R = 160 cm during time interval τ = 10.0 s. Calculate the following quantities averaged over that time:

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(a) the mean velocity $\rm \big\langle v \big\rangle ;$

(b) the modulus of mean velocity vector $\rm | \big\langle v \big\rangle | ;$

(c) the modulus of mean vector of the total acceleration $\rm | \big\langle w \big\rangle |$ if the point moved with constant tangent acceleration.​

Answer 1

(a) mean velocity

⟨v⟩ = $\sf \dfrac{total\:distance\:covered}{time\:ellapsed}$

$\tt = \dfrac{s}{t}$

$\tt = 50\: cm/s \:\;\;\;\;\; (1)$

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(b) modulus of mean velocity vector

$\tt | \langle \vec{v} \rangle | = \dfrac{\Delta |\vec{r}|}{\Delta t} = \dfrac{2R}{\tau}$

$\tt = 32\: cm/s \:\;\;\:\;\; (2)$

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(c) Let the point moves from i to f along the half circle (Fig.) and v_0 and v be the space at the point respectively.

$\tt \sf{we\: have} \:\:\:\;\;\;\;\;\; \dfrac{dv}{dt} = w_t$

$\tt \langle v \rangle = \left( \dfrac{\displaystyle\int_{0}^{t} \tt (v_0 + w_t t)dt}{\displaystyle\int_{0}^{t} \tt dt} \right)$

$\tt = \dfrac{v_0+v}{2}$ ....(3)

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so from 1&3,

$\tt \dfrac{v_0+v}{2} = \dfrac{\pi R}{\tau}$ ....(4)

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$\tt | \langle \vec{v} \rangle | = \dfrac{2\pi R}{\tau^2}$