Net work

Net work W is equal to change in kinetic energy . So W = 0.5 * 4* (61 - 34) = 54 J

Some work must be done in order to change the velocity of the body.The work done is computed by subtracting the final Kinetic Energy of the body by Initial Kinetic Energy(This difference gives us the work done on the vehicle).

${ W }_{ net }=\quad \Delta K.E\quad ={ \quad K.E }_{ final }-{ K.E }_{ initial }$

Kinetic Energy of a body is given by: $K.E = \frac { 1 }{ 2 } mv^2$

In the question, the velocity is given in the form of vectors.The magnitude of the velocity of the x and y component together is given by $v\quad =\sqrt { v_{ x }^{ 2 }+v_{ y }^{ 2 } }$ .We need only the magnitude because Energy is a scalar quantity and we need only the magnitude of the velocity to compute the Kinetic Energy of the body.

Computation: m = 4
${ V }_{ initial } = -5\hat { i } -3\hat { j } = \sqrt { -5^{ 2 }+-3^{ 2 } } = \sqrt {34}$

${ V }_{final } = 5\hat { i } + 6\hat { j } = \sqrt { 5^{ 2 }+6^{ 2 } } = \sqrt {61}$

$K.E_{initial} = 1/2 * 4 * {\sqrt {34}}^2 = 2 * 34 = 68$

$K.E_{final} = 1/2 * 4 * {\sqrt {61}}^2 = 2 * 61 = 122$

${ W }_{ net }= 122 - 68 = 54J$

net work done = change in kinetic energy of the body= 1/2(61-34)=54j

Here we assume that , there is no change in potential energy thn calculate work done as change in kinetic energy ''''''''''''''''''''''''''

yup, its the kinetic energy formula