What is the comet's speed at its perihelion

[SABRINA]

The orbit of Halley's Comet around the Sun is a long thin ellipse. At its aphelion (point farthest from the Sun), the comet is 5.6 10^12 m from the Sun and moves with a speed of 13.0 km/s. What is the comet's speed at its perihelion (closest approach to the Sun) where its distance from the Sun is 8.4 10^10 m?

 

[mathwonk]

i think i would look up the three laws governing this movement, what are they called?

 

[BobG]

The orbit of Halley's Comet around the Sun is a long thin ellipse. At its aphelion (point farthest from the Sun), the comet is 5.6 10^12 m from the Sun and moves with a speed of 13.0 km/s. What is the comet's speed at its perihelion (closest approach to the Sun) where its distance from the Sun is 8.4 10^10 m?

There's a couple of ways to figure this out: Conservation of energy and conservation of momentum.

Because the comet's velocity is perpendicular to the radius at perigee and apogee, using the angular momentum is the easiest way. Specific angular momentum is equal to the radius times the speed times the sine of the angle between the two vectors (so is angular momentum, but just looking at the specific angular momentum per unit of mass is good enough for this).

$$h=r v sin \theta$$

Or, since the angle at both apogee and perigee are 90 degress,:

$$r_a v_a = r_p v_p$$

You can also find this based on conservation of energy. Total energy equals kinetic plus potential energy. Total energy stays constant - while kinetic energy and potential energy change depending where in the orbit that the comet is. Once again, you could use specific energy per unit of mass.

$$\frac{1}{2} v_a^2 - \frac{GM}{r_a}=\frac{1}{2} v_p^2 - \frac{GM}{r_p}$$
(Gravity pulls 'down' towards the center of the Sun, hence the negative sign in front of potential energy.)

You should get the same answer both ways.