# Math Help - gravity force

1. ## gravity force

1>A system consists of three particles, each of mass 5.00 g, located at the corners of an equilateral triangle with sides of 30.0 cm.
(a) Calculate the potential energy of the system.
(b) If the particles are released simultaneously, where will they collide?

2>At the Earth’s surface a projectile is launched straight up at a speed of 10.0 km/s. To what height will it rise? Ignore air resistance and the rotation of the Earth

2. Originally Posted by gracy
1>A system consists of three particles, each of mass 5.00 g, located at the corners of an equilateral triangle with sides of 30.0 cm.
(a) Calculate the potential energy of the system.
(all quantities will be converted to their SI equivalents in what follows)

The potential energy of this system is equal to the work required to remove
the points to infinity. This can be done in a number of ways, but the
result is always the same, and equal to:

$
E=3 \int_{r=0.3}^{\infty} \frac{G\,m_1\,m_2}{r^2}\, dr=3\,\frac{G\,m_1\,m_2}{0.3}
$

but $m_1=m_2=0.005 kg$ and $G\approx 6.67\ 10^{-11} \rm{Nm^3/kg^2}$, so:

$E\approx 1.67\ 10^{-14}\mbox{joules}$

(b) If the particles are released simultaneously, where will they collide?
The centre of mass of a free system like this remains at rest as long as there are
no external forces, so the particles must collide at their centre of mass, which is
at the centre of the equilateral triangle.

RonL

3. Originally Posted by gracy
2>At the Earth’s surface a projectile is launched straight up at a speed of 10.0 km/s. To what height will it rise? Ignore air resistance and the rotation of the Earth
We have the projectiles acceleration:

$a=-g$

so integrating once gives its velocity:

$v=-gt +c$,

but at $t=0\ v=10\ \rm{m/s}$ so $c=10$.

Integrate again to get its displacement:

$d=-gt^2/2+10t +k$,

but at $t=0\ d=0\ \rm{m}$ so $k=0$.

Now the maximum height is reached when $v=0$ which is at $t=10/g\ \rm{s}$, so the maximum height reached is:

$d_{max}=-50/g+100/g=150/g\ \rm{m}$.

and as $g \approx 9.81 \ \rm{m/s^2}$,

$d_{max}\approx 15.3 \ \rm{m}$

RonL