Linear momentum of particle is increased by (a)100% (b)1% Without changing it's mass. Find percentage increase in its kinetic energy in both cases?

Question

Answer

Answer

A)Let initial momentum be p
then KE = p^2/2m
when momentum is increased by 100% then
p'=2p
hence KE'=4p^2/2m
so percentage increase in KE =[(4p^2/2m - p ^2/2m)/p^2/2m]×100
=300
b) let initial momentum be p
KE=p^2/2m
when momentum is increased by 1% then
p'=101p/100
KE'=(1.01p)^2/2m
% increase in KE= 2.01

Answer

Linear Momentum and Kinetic Energy

Linear momentum is a vector quantity that measures the motion of an object in a straight line. It is defined as the product of an object's mass and its velocity. Mathematically, linear momentum (p) is given by the equation:

p = m * v

where p is the linear momentum, m is the mass of the object, and v is its velocity.

On the other hand, kinetic energy is the energy possessed by an object due to its motion. It is given by the equation:

KE = (1/2) * m * v^2

where KE is the kinetic energy, m is the mass of the object, and v is its velocity.

Increasing Linear Momentum by 100%

When the linear momentum of a particle is increased by 100% without changing its mass, the new linear momentum (p') will be twice the original momentum (p).

p' = 2 * p

To find the percentage increase in kinetic energy, we need to calculate the new kinetic energy (KE') and compare it with the original kinetic energy (KE).

Since the mass remains constant, the new kinetic energy will be:

KE' = (1/2) * m * (2v)^2
= 2 * (1/2) * m * v^2
= 2 * KE

The new kinetic energy is twice the original kinetic energy. Therefore, the percentage increase in kinetic energy is given by:

Percentage Increase = ((KE' - KE) / KE) * 100
= ((2 * KE - KE) / KE) * 100
= 100%

So, when the linear momentum of a particle is increased by 100%, the kinetic energy increases by 100%.

Increasing Linear Momentum by 1%

When the linear momentum of a particle is increased by 1% without changing its mass, the new linear momentum (p') will be slightly greater than the original momentum (p).

p' = p + (1/100) * p
= (101/100) * p

To find the percentage increase in kinetic energy, we need to calculate the new kinetic energy (KE') and compare it with the original kinetic energy (KE).

Since the mass remains constant, the new kinetic energy will be:

KE' = (1/2) * m * ((101/100) * v)^2
= (1/2) * m * (10201/10000) * v^2
= (51/50) * KE

The new kinetic energy is slightly greater than the original kinetic energy. Therefore, the percentage increase in kinetic energy is given by:

Percentage Increase = ((KE' - KE) / KE) * 100
= (((51/50) * KE - KE) / KE) * 100
= (1/50) * 100
= 2%

So, when the linear momentum of a particle is increased by 1%, the kinetic energy increases by 2%.

Conclusion

In summary, when the linear momentum of a particle is increased by 100% without changing its mass, the kinetic energy increases by 100%. On the other hand, when the linear momentum is increased by 1%, the kinetic energy increases by 2%. This relationship between momentum and kinetic energy highlights

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