Question

if the linier momentum is increased by 50 percent the kinetic energy will increased by​

Answer 1

SoLuTioN :

⏭ Given:

✏ The linear momentum of a body is increased by 50%

⏭ To Find:

✏ The kinetic energy is increased by...

⏭ Relation:

✏ Relation between linear momentum and kinetic energy is given by

$\orange{ \bigstar} \: \underline{ \boxed{ \bold{ \rm{ \pink{K = \frac{ {P}^{2} }{2m} }}}}} \: \orange{ \bigstar}$

⏭ Calculation:

✏ As per relation, it is clear that

$\star \: \underline{\underline{ \sf{ \red{K \propto {P}^{2} }}}} \: \star$

✏ Here m remains constant.

✏ When the momentum of a body is increased by 50%, its momentum will become

$\implies \sf \: P_2 = P_1 + \frac{50}{100} P_1 = (1.5) P_1 \\ \\ \therefore \sf \: \frac{K_2}{K_1} = \frac{ {P_2}^{2} }{ {P_1}^{2} } = {1.5}^{2} \\ \\ \therefore \sf \: K_2 = (2.25)K_1$

✏ Percentage increase in the kinetic energy of the body

$\implies \sf \: \frac{K_2 - K_1}{K_1} \times 100 = \frac{(2.25)K_1 - K_1}{K_1} \times 100 \\ \\ \implies \sf \: \underline{ \boxed{ \bold{ \sf{ \purple{\% \frac{ \triangle{K}}{K} = 125\%}}}}} \: \red{ \bigstar}$

Answer 2

Solution:

Let us assume the initial velocity before increasing to be 'u' and the final velocity after increasing Momentum to be 'v'

According to the question, Momentum is increased by 50%.

→ mv = mu + 1/2 mu

→ mv = 1.5 mu

→ v = 1.5u  [Since Mass is constant]

Now new velocity after increasing is 1.5 u

→ New Kinetic Energy = 1/2 × m × (1.5u)²

→ New Kinetic Energy = 1/2 × m × 2.25 u²

→ New Kinetic energy = 2.25 ( 1/2 mu² ) = 2.2.5 ( K.E )

Difference in Kinetic Energies:

→ 2.25 ( K.E ) - K.E)

→ 1.25 K.E.

Hence Increase in Kinetic Energy is given as:

$\boxed{ Increase\: in \: K.E.\: = \dfrac{ Increase\:in\:K.E. }{ Initial\:K.E.} \times 100}\\\\\\\implies \dfrac{1.25\:\:K.E.}{K.E.}\times 100\:\: = 125 \%$

Hence Kinetic Energy is increased by 125%.