Question

2) By what factor does an object's kinetic energy change if its speed is doubled? By what factor does its momentum change?

Answer 1

When an object's speed is doubled, its kinetic energy increases by a factor of four while its momentum doubles. In elastic collisions, both momentum and kinetic energy are conserved, whereas in inelastic collisions, only momentum is conserved with kinetic energy being converted into other energy forms.

Step-by-step explanation:

Kinetic Energy and Momentum Changes

When an object's speed is doubled, its kinetic energy changes by a factor of four. This is because kinetic energy is proportional to the square of the speed (KE = 1/2 m v²). Therefore, if the velocity (v) is doubled, the kinetic energy increases by 2², which is 4 times the original kinetic energy.

As for momentum, it changes linearly with velocity (p = m v). If the speed of the object is doubled, the momentum is simply doubled.

When two objects collide:

In an inelastic collision, momentum is conserved, but kinetic energy is not; it is often converted into other forms of energy, such as heat or sound.

For objects with the same momentum, the object with smaller mass has a larger kinetic energy. Conversely, for objects with the same kinetic energy, the object with the larger mass will have a larger momentum.

Answer 2

Let the object has mass m and speed v initially.

Then, kinetic energy will be

`K\mathrm{}E\text{. =}(1)/(2)mv^2`

and the momentum will be

`p=mv`

Now, if the speed is doubled, v'=2v

then kinetic energy will be

`\begin{gathered} K\mathrm{}E\mathrm{}^(\prime)=(1)/(2)m*4v^2 \\ =4* K.E. \end{gathered}`

Also, momentum will be

`\begin{gathered} p^(\prime)=m*2v \\ =2p \end{gathered}`

Hence, kinetic energy changes by a factor of 4 and momentum changes by a factor of 2 when speed is doubled.