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PhMichael
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I have a body of mass m, which is moving in a circular path around some planet. At a certain instant, the object explodes into two equal bodies. I'm given that the tangential velocity doesn't change as a result of the explosion, in addition, the system's kinetic energy increases by a factor of k (k>1). I'm asked to determine the minimal radius (measured from the binding center) of the two half-bodies.
2. The attempt at a solution
The total energy before the explosion occurred is: (M=the mass of the planet)
[tex] E_{tot} = \frac{1}{2}mv_{\theta}^{2} - \frac{GMm}{R} [/tex]
Now, the total energy after the explosion:
[tex] E_{tot} = \frac{k}{2}mv_{\theta}^{2} - \frac{GMm}{R_{min}} [/tex]
now, we can obtain the tangential velocity from Newton's II law:
[tex] -\frac{GMm}{R^2} = -m \frac{v_{\theta}^{2}}{R} [/tex]
[tex]v_{\theta}^{2} = \frac{GM}{R}[/tex]
equating both expressions of [tex]E_{tot}[/tex] as the total energy is conserved and using the last relation, yields:
[tex]-\frac{GMm}{2R} = \frac{kGMm}{2R}-\frac{GMm}{R_{min}}[/tex]
therefore,
[tex]R_{min} = \frac{2R}{1+k}[/tex]
The right answer, however, is:
[tex]R_{min} = \frac {R}{1+\sqrt{k-1}}[/tex]
what have I done wrong?
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Homework Statement
I have a body of mass m, which is moving in a circular path around some planet. At a certain instant, the object explodes into two equal bodies. I'm given that the tangential velocity doesn't change as a result of the explosion, in addition, the system's kinetic energy increases by a factor of k (k>1). I'm asked to determine the minimal radius (measured from the binding center) of the two half-bodies.
2. The attempt at a solution
The total energy before the explosion occurred is: (M=the mass of the planet)
[tex] E_{tot} = \frac{1}{2}mv_{\theta}^{2} - \frac{GMm}{R} [/tex]
Now, the total energy after the explosion:
[tex] E_{tot} = \frac{k}{2}mv_{\theta}^{2} - \frac{GMm}{R_{min}} [/tex]
now, we can obtain the tangential velocity from Newton's II law:
[tex] -\frac{GMm}{R^2} = -m \frac{v_{\theta}^{2}}{R} [/tex]
[tex]v_{\theta}^{2} = \frac{GM}{R}[/tex]
equating both expressions of [tex]E_{tot}[/tex] as the total energy is conserved and using the last relation, yields:
[tex]-\frac{GMm}{2R} = \frac{kGMm}{2R}-\frac{GMm}{R_{min}}[/tex]
therefore,
[tex]R_{min} = \frac{2R}{1+k}[/tex]
The right answer, however, is:
[tex]R_{min} = \frac {R}{1+\sqrt{k-1}}[/tex]
what have I done wrong?
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