An observer O₁ standing on ground finds that momentum of a projectile of mass 2 kg changes with time as ${\vec{P}}_{01} = (4 \hat{i} + 20 \hat{k}) kgm / s$ Acceleration due to gravity is $\vec{g} = (10 \hat{k}) m / s^{2}$ and there is a wind blowing in horizontal direction. Another observer O₂ driving a car observes that momentum of the same projectile changes with time as
${\vec{P}}_{02} = (8 \hat{i} - 32 t \hat{j} + 20 \hat{k}) kgm / s .$
celeration of the car at $t = \frac{1}{8} s$ in m/ $s^{2}$

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$- 2 \hat{i} + 2 \hat{j}$

$4 \hat{i} + 2 \hat{j}$

-4 $\hat{i} - 2 \hat{j}$

$8 \hat{i}$

answer is A.

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Detailed Solution

For O₁force on the projectile is
${\vec{F}}_{01} = \frac{\vec{d} p_{01}}{dt} = (4 \hat{i} + 20 \hat{k}) N$
his is real force on the projectile due to gravity and blowing wind. For O₂ force on the projectile is –
${\vec{F}}_{02} = \frac{\vec{d} p_{02}}{dt} = (8 \hat{i} - 32 t \hat{j} + 20 \hat{k}) N {\vec{F}}_{real} + {\vec{F}}_{pseudo} = 8 \hat{i} - 32 t \hat{j} + 20 \hat{k} (4 \hat{i} + 20 \hat{k}) + {\vec{F}}_{pseudo} = 8 \hat{i} - 32 t \hat{j} + 20 \hat{k} ∴ {\vec{F}}_{pseudo} = 4 \hat{i} - 32 t \hat{j} At = \frac{1}{8} s {\vec{F}}_{pseudo} = 4 \hat{i} - 4 \hat{j} ∴ - m {\vec{a}}_{car} = 4 \hat{i} - 4 \hat{j} {\vec{a}}_{car} = - 2 \hat{i} + 2 \hat{j}$