**Relation between Momentum and K.E**

**Question:**

If Momentum of a body is increased by 20%, what is percentage change in the K.E of the body?

a) 40%

b) 44%

c) 20%

d) 10%

**Answer:**

44%

**Solution:**

Let the original momentum of the body be P, Then:

P=mv

Momentum is increased by 20%, so new momentum will be:

P' = P + 20%P

P' = P + 0.2P = 1.2P

K.E = 0.5 mv^2

=0.5 m^2v^2/m (multiply and divide by m)

K.E = 0.5 P^2 / m

New K.E = K.E'

= 0.5 (P')^2/m

= 0.5 (1.2P)^2 / m

= 0.5 (1.44) P^2 / m

= 1.44 K.E

Percentage Change in K.E = (New K.E - Old K.E) /Old K.E x 100%

= (1.44K.E - K.E)/K.E x 100%

= 44%

Momentum of a body is given by :

P = mv

P can be varied by varying either m or v. Since mass of a body is a constant quantity so we can change momentum of a body by changing its velocity.

Momentum will be increased by 20% if the velocity is changed by 20%, since its a linear relation between the two.

Let

mass = m = 10kg

velocity = v = 100m/s

Momentum with these values = P = 1000 kgm/s

K.E with these values =

After 20% increase velocity becomes = v' = 120m/s

so new momentum = 1200 kgm/s

Change in momentum = (1200-1000)/1000 x 100%

= 0.2 x 100%

= 20%

Change in K.E = (72000 - 50000)/50000 x 100%

= 0.44 x 100%

= 44%

Which agrees with the above result.

Formula to be used for finding the percentage change:

Percentage Change = New Value - Old value/Old value x 100%

In case of increment the change is positive.

In case of decrement the change is negative.

K.E = 0.5 P^2 / m

**---------- (i)**New K.E = K.E'

= 0.5 (P')^2/m

= 0.5 (1.2P)^2 / m

= 0.5 (1.44) P^2 / m

= 1.44 K.E

**(using i)**Percentage Change in K.E = (New K.E - Old K.E) /Old K.E x 100%

= (1.44K.E - K.E)/K.E x 100%

= 44%

**Second Method:**Momentum of a body is given by :

P = mv

P can be varied by varying either m or v. Since mass of a body is a constant quantity so we can change momentum of a body by changing its velocity.

Momentum will be increased by 20% if the velocity is changed by 20%, since its a linear relation between the two.

Let

mass = m = 10kg

velocity = v = 100m/s

Momentum with these values = P = 1000 kgm/s

K.E with these values =

**K.E = 50000 J**After 20% increase velocity becomes = v' = 120m/s

so new momentum = 1200 kgm/s

Change in momentum = (1200-1000)/1000 x 100%

= 0.2 x 100%

= 20%

**New K.E =****72000 J**Change in K.E = (72000 - 50000)/50000 x 100%

= 0.44 x 100%

= 44%

Which agrees with the above result.

Formula to be used for finding the percentage change:

Percentage Change = New Value - Old value/Old value x 100%

In case of increment the change is positive.

In case of decrement the change is negative.

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