A car moves in positive y - direction with velocity v proportional to distance travelled y as v(y) propto yβ, where β is a positive constant. The car covers a distance L with average velocity proportional to L as langle v propto L1 / 3. The constant β is given as

Question

A car moves in positive y - direction with velocity v proportional to distance travelled y as v(y) propto yβ, where β is a positive constant. The car covers a distance L with average velocity < v > proportional to L as langle v> propto L1 / 3. The constant β is given as (A) (1/4) (B) (1/3) (C) (2/3) (D) (1/2)

Tardigrade Admin · Accepted Answer

Relation between average velocity and instantaneous velocity is

< v >= \frac{1}{t _{2} - t _{1}} \int_{t_{1}}^{t_{2}} v (t) d t

In the given problem,

v (t) \propto y^{β}

(independent of

t

)
So,

⟨ v ⟩ \propto v (t) or (v) \propto y^{β}

But according to question,

< v >\propto y^{1/3}

So,

β = \frac{1}{3}

Q. A car moves in positive $y$ - direction with velocity $v$ proportional to distance travelled $y$ as $v (y) \propto y^{β}$ , where $β$ is a positive constant. The car covers a distance $L$ with average velocity $< v >$ proportional to $L$ as $⟨ v >\propto L^{1/3}$ The constant $β$ is given as

$\frac{1}{4}$

$\frac{1}{3}$

$\frac{2}{3}$

$\frac{1}{2}$

Q. A car moves in positive y - direction with velocity v proportional to distance travelled y as v(y)∝yβ, where β is a positive constant. The car covers a distance L with average velocity <v> proportional to L as ⟨v>∝L1/3 The constant β is given as

41​

31​

32​

21​

Relation between average velocity and instantaneous velocity is <v>=t2​−t1​1​∫t1​t2​​v(t)dt In the given problem, v(t)∝yβ (independent of t ) So, ⟨v⟩∝v(t) or (v)∝yβ But according to question, <v>∝y1/3 So, β=31​

$\frac{1}{4}$

$\frac{1}{3}$

$\frac{2}{3}$

$\frac{1}{2}$