Question

A particle moves with a speed v = kt² , where k is a constant and t is time, along a straight line. The change in its speed in time t=0 to t= T is
A) KT² / 3
B) KT² / 4
C) KT² / 2
D) KT²
​

Answer 1

Change in its speed = V = ds/dt

Solution:

$\implies\rm v = kt^2$

$\implies \rm \dfrac{ds}{dt} = kt^2$

$\implies\rm ds = kt^2(dt)$

$\implies\rm \displaystyle\int ds =\displaystyle \int ^T _0 \dfrac{1}{3}kt^3$

$\implies\rm s = \dfrac{k(T^3-0^3)}{3}$

$\implies \bf s = \dfrac{kT^3}{3}$

$\to \rm v = \dfrac{ds}{dt} = \dfrac{kT^3}{3(T-0)}$

$\implies\bf v = \dfrac{kT^2}{3}$

option A) is correct.

Answer 2

Explanation:

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Change in its speed = V = ds/dt

Solution:

$\implies\rm v = kt^2$

$\implies \rm \dfrac{ds}{dt} = kt^2$

$\implies\rm ds = kt^2(dt)$

$\implies\rm \displaystyle\int ds =\displaystyle \int ^T _0 \dfrac{1}{3}kt^3$

$\implies\rm s = \dfrac{k(T^3-0^3)}{3}$

$\implies \bf s = \dfrac{kT^3}{3}$

$\to \rm v = \dfrac{ds}{dt} = \dfrac{kT^3}{3(T-0)}$

$\implies\bf v = \dfrac{kT^2}{3}$

option A) is correct.

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