# Question #90693

Jan 31, 2017

Recall the Newton' second Law of motion for an object of mass $m$ acted upon a force $\vec{F}$
$\vec{F} = m \vec{a}$ ......(1)
where $\vec{a}$ is the acceleration produced.

If a force acts through a displacement $\vec{s}$,
Work done $W = \vec{F} \cdot \vec{s}$ .....(2)
Using (1) we can write (2) as
$W = m \vec{a} \cdot \vec{s}$ ....(3)

Recall the kinematic equation
${\vec{v}}^{2} - {\vec{u}}^{2} = 2 \vec{a} \cdot \vec{s}$ .....(4)
where $\vec{u}$ is the initial velocity, $\vec{v}$ is the final velocity.

Using (4) equation (3) becomes*
$W = \frac{1}{2} m \left({\vec{v}}^{2} - {\vec{u}}^{2}\right)$
Inserting given values
$W = \frac{1}{2} \times 8.1 \left({\left(9.6 \hat{i} + 2.3 \hat{j}\right)}^{2} - {\left(9.9 \hat{i} + 6.0 \hat{j}\right)}^{2}\right)$
We know that square of a vector is dot product of vector with itself hence, above equation becomes
$W = \frac{1}{2} \times 8.1 \left(\left({9.6}^{2} + {2.3}^{2}\right) - \left({9.9}^{2} + {6.0}^{2}\right)\right)$
$\implies W = - 148.1 J$

*recall that equation is $W = \left(\text{Final kinetic energy"-"Initial Kinetic energy}\right)$