A body of mass m breaks up into 2 parts of rest mass m1 and m2 with velocity v1 and v2 ,what will be the energies of m1 and m2 in terms of m,m1,m2 and c,the velocity of light.
To find the energies of m1 and m2, we can use the concept of relativistic energy.
The relativistic energy of an object can be calculated using the equation:
E = γmc^2
Where E is the energy, m is the rest mass of the object, c is the speed of light in a vacuum, and γ (gamma) is the Lorentz factor.
The Lorentz factor γ is given by the equation:
γ = 1 / √(1 - (v/c)^2)
Where v is the velocity of the object and c is the speed of light.
Now, let's calculate the energies of m1 and m2 separately.
For m1:
E1 = γ1 * m1 * c^2
To find γ1, we need to calculate the velocity of m1. Since the total momentum before the breakup is equal to the total momentum after the breakup, we can use the conservation of momentum:
mv = m1v1 + m2v2
Given that the velocity of the two parts after the breakup is v1 and v2, respectively, and assuming the initial velocity v is zero, we get:
0 = m1v1 + m2v2
From this equation, we can solve for v1:
v1 = - (m2v2) / m1
Now, substitute this value of v1 into the equation for γ1:
γ1 = 1 / √(1 - ((-(m2v2) / m1) / c)^2)
Simplifying further:
γ1 = 1 / √(1 - ((m2v2)^2 / (m1^2 * c^2)))
Now, substitute γ1 into the equation for E1:
E1 = (1 / √(1 - ((m2v2)^2 / (m1^2 * c^2)))) * m1 * c^2
For m2, repeat the same steps with the appropriate variables and velocity:
v2 = - (m1v1) / m2
γ2 = 1 / √(1 - ((m1v1)^2 / (m2^2 * c^2)))
E2 = (1 / √(1 - ((m1v1)^2 / (m2^2 * c^2)))) * m2 * c^2
So, the energy of m1 is given by E1 = (1 / √(1 - ((m2v2)^2 / (m1^2 * c^2)))) * m1 * c^2, and the energy of m2 is given by E2 = (1 / √(1 - ((m1v1)^2 / (m2^2 * c^2)))) * m2 * c^2.