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  4. A particle of mass 0.01 kg travels along a space curve with velocity given by 4 hat i +16 hat k m / s. After some time, its velocity becomes 8 hat i +20 hat j m / s due to the action of a conservative force. The work done on the particle during this interval of time is :

Q. A particle of mass $0.01 \,kg$ travels along a space curve with velocity given by $4 \hat{ i }+16 \hat{ k }\, m / s$. After some time, its velocity becomes $8 \hat{ i }+20 \hat{ j } \,m / s$ due to the action of a conservative force. The work done on the particle during this interval of time is :

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Solution:

Given mass and initial velocity
$m=0.01 \,kg$
$\vec{ u }=4 \hat{ i }+16 \hat{ k } m / s$
$|\vec{ u }| =\sqrt{(4)^{2}+(16)^{2}} $
$|\vec{ u }| =\sqrt{272} \,m / s $
$ \vec{ v } =8 \hat{ i }+20 \hat{ j } $
$|\vec{ v }| =\sqrt{(8)^{2}+(20)^{2}} $
$|\vec{ v }| =\sqrt{464}\, m / s $
$ W =\vec{ F } \cdot \vec{ s } $
$ W =m \vec{ a } \cdot \vec{ s }$
$W=m\left(\frac{v^{2}-u^{2}}{2}\right)$
$W=0.01 \times\left[\frac{464-272}{2}\right]$
$W=0.01 \times 96$
$W=0.96\, J$

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