Momentum and MassIs Mass a Relative Quantity???

By Connie Wong

Momentum

Classical momentum:

p = mov

For speeds much less than the speed of light

Relativistic momentum:

p = mov/ (√1- v2/c2)

For speeds closer to the speed of light

Mass

p = mov/ (√1- v2/c2)

The equation of relativistic momentum

suggests a relativistic interpretation of mass

Mass is Relative!

Mass Increase Formula

The mass of an object is measured to

increase as its speed increases:

m = mo/ (√1- v2/c2)

mo = the rest mass of the object, the mass it has as measured in a reference frame in which it is at rest

m = the relativistic mass, the mass it will be measured to have in a reference frame in which it moves at speed v

Whiteboards:

1. Calculate the mass of an electron when it has a speed of a) 4.00E7 m/s in the CRT of a TV set, and b) 0.98c in an accelerator used for cancer therapy.

mo of an electron: 9.11 E-31kg

m = mo/ (√1- v2/c2)

1. Calculate the mass of an electron when it has a speed of a) 4.00E7 m/s in the CRT of a TV set, and b) 0.98c in an accelerator used for cancer therapy.

a) v = 4.00E7m/s, mo = 9.11E-31kg, c = 3.00E8m/s

m = mo / (√1- v2/c2)

m = 9.11E-31 / (√1- (4.00E7)2/ (3.00E8)2

m = 9.19E-31 kg

b) v = 0.98c, mo = 9.11E-31kg, c = 3.00E8m/s

m = mo/ (√1- (0.98c)2/ c2)

m = 5.0mo = Five times its rest mass!

m = 4.56E-30kg

More Whiteboards:

2. What is the mass of a proton traveling at v = 0.75c?

Rest mass of proton= 1.673E-27kg

m = mo/ (√1- v2/c2)

m = 1.673E-27/ (√1- (0.75c)2/c2)

m = 2.53E-27kg

3. At what speed v will the mass of an object be 15 percent greater than its rest mass?

m = mo/ (√1- v2/c2)

v = ?

1.15mo = mo/ (√1- v2/c2)

1.15mo* (√1- v2/c2)= mo ...mo cancels...

√1- v2/c2 = 1/1.15 ...solve for v...

v = 1.48E8m/s or 0.494c

The Ultimate Speed

m = mo/ (√1- v2/c2)

As v increases, the mass of the object becomes larger and larger.

If v = c, the denominator would equal zero and mass would be infinite.

To accelerate an object up to v = c would require infinite energy, which is impossible.

Thus: the speed of an object cannot equal or exceed the speed of light.

The End