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A particle of mass m is fired upwards from the surface of a planet of mass M
and radius R with velocity
Show that the maximum height the particle attains is R/3.

Respuesta :

onyebuchinnaji

At the maximum height the particles position has been shown as [tex]\frac{R}{3}[/tex].

The given parameters;

  • mass of the particle, = M
  • radius of the particle, = R
  • maximum height the particle attains = R/3

The velocity of the particle is given as;

[tex]v^2 = \frac{GM}{2R}[/tex]

The total mechanical energy of the particle is given as;

[tex]-\frac{GMm}{R} + \frac{1}{2} mv^2 = - \frac{GMm}{R} + \frac{1}{2} mv^2[/tex]

At maximum height the final velocity of the particle is zero.

[tex]-\frac{GMm}{R} + \frac{1}{2} mv^2 = - \frac{GMm}{R +h}[/tex]

[tex]-\frac{GMm}{R} + \frac{1}{2} m\times \frac{GM}{2R} = - \frac{GMm}{R +h}\\\\-\frac{GMm}{R} + \frac{GMm}{4R} = - \frac{GMm}{R +h}\\\\- \frac{3GMm}{4R} = - \frac{GMm}{R +h}\\\\- \frac{3}{4R} = - \frac{1}{R+ h} \\\\3(R+ h) = 4R\\\\3R + 3h = 4R\\\\3h = 4R - 3R\\\\3h = R\\\\h = \frac{R}{3}[/tex]

Thus, at the maximum height the particles position has been shown as [tex]\frac{R}{3}[/tex].

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