If two different masses have the same kinetic energy, their momenta are:
a. proportional to the squares of their masses
b. proportional to their masses
c. proportional to the square roots of their masses
d. inversely proportional to their masses
e. inversely proportional to the square roots of their masses
I am torn between b and c.
Would it be choice b.?
To determine which option is correct, we need to relate kinetic energy and momentum.
The formula for kinetic energy (KE) is:
KE = (1/2)mv^2
where m represents the mass of an object and v represents its velocity.
The formula for momentum (p) is:
p = mv
where m is the mass of an object and v is its velocity.
Given that two different masses have the same kinetic energy, we can set up the following equation:
(1/2)m1v1^2 = (1/2)m2v2^2
Since both kinetic energies are the same, we can cancel out the (1/2) factor:
m1v1^2 = m2v2^2
Now, let's analyze the options:
a. Proportional to the squares of their masses
If momentum were proportional to the squares of the masses, our equation would be:
(m1)^2v1^2 = (m2)^2v2^2
This is different from the equation above, so option (a) is not correct.
b. Proportional to their masses
If momentum were proportional to the masses, our equation would be:
m1v1^2 = m2v2^2
This aligns with the equation we derived, so option (b) could be the correct answer.
c. Proportional to the square roots of their masses
If momentum were proportional to the square roots of the masses, our equation would be:
√(m1)v1^2 = √(m2)v2^2
This is different from the equation we derived, so option (c) is not correct.
d. Inversely proportional to their masses
If momentum were inversely proportional to the masses, our equation would be:
v1^2/m1 = v2^2/m2
This is different from the equation we derived, so option (d) is not correct.
e. Inversely proportional to the square roots of their masses
If momentum were inversely proportional to the square roots of the masses, our equation would be:
v1^2/√(m1) = v2^2/√(m2)
This is different from the equation we derived, so option (e) is not correct.
Based on the analysis above, the correct answer is option (b), which states that momenta are proportional to their masses when two different masses have the same kinetic energy.