How do I calculate the new speed given a doubled momentum?

[Peter G.]

Hi,

I was given a question in which I had to work out the speed of an object given its momentum and its mass.

Now I had to answer what would be the new speed if the momentum doubled.

So, from what I understand, the rest mass can't change, it is like a constant for a given body. So, in other words, in this case, it is the velocity that is going to change.

In order to find by what factor v must be increased in order to yield an increase in momentum of two I tried to rearrange the following equation:

v/√1-(v²/c² = 2

Is this what I should do?

Can anyone please maybe give me a hint of what step I should take?

 

[Curious3141]

Hi,

I was given a question in which I had to work out the speed of an object given its momentum and its mass.

Now I had to answer what would be the new speed if the momentum doubled.

So, from what I understand, the rest mass can't change, it is like a constant for a given body. So, in other words, in this case, it is the velocity that is going to change.

In order to find by what factor v must be increased in order to yield an increase in momentum of two I tried to rearrange the following equation:

v/√1-(v²/c² = 2

Is this what I should do?

Can anyone please maybe give me a hint of what step I should take?

The formula for the relativistic momentum is $p = {\gamma}mv$, where $\gamma =\frac { 1 }{ \sqrt { 1-\frac { { v }^{ 2 } }{ { c }^{ 2 } } } }$, which gives $p = \frac { mv }{ \sqrt { 1-\frac { { v }^{ 2 } }{ { c }^{ 2 } } } }$

You're given that $p_{new} = 2p_{old}$.

So what you need to do is to solve this equation:

$$\frac { mv }{ \sqrt { 1-\frac { { v }^{ 2 } }{ { c }^{ 2 } } } } = 2\frac { mu }{ \sqrt { 1-\frac { { u }^{ 2 } }{ { c }^{ 2 } } } }$$

where v is the new speed and u is the old speed (which you've already calculated). Solve for v.