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QuestionHalley's comet moves about the sun in an elliptical orbit, with its closest approach to the Sun being 59 A.U. and its greatest distance being 35 A.U. . Note one A.U. is the Earth's Sun Distance. If the speed at closest approach is 54 km/s54 \mathrm{~km} / \mathrm{s}, find the speed when furthest from the Earth.

Studdy Solution

STEP 1

1. Halley's comet follows an elliptical orbit around the Sun.
2. The closest distance (perihelion) to the Sun is 59 59 A.U.
3. The farthest distance (aphelion) from the Sun is 35 35 A.U.
4. The speed of the comet at perihelion is 54 km/s 54 \, \text{km/s} .
5. We need to find the speed of the comet at aphelion.

STEP 2

1. Use the conservation of angular momentum.
2. Set up the equation for angular momentum at perihelion and aphelion.
3. Solve for the speed at aphelion.

STEP 3

Recall the principle of conservation of angular momentum, which states that the angular momentum of the comet is constant throughout its orbit. The angular momentum L L is given by:
L=m⋅v⋅r L = m \cdot v \cdot r
where m m is the mass of the comet, v v is the velocity, and r r is the distance from the Sun.

STEP 4

Set up the equation for angular momentum at perihelion and aphelion:
m⋅vperihelion⋅rperihelion=m⋅vaphelion⋅raphelion m \cdot v_{\text{perihelion}} \cdot r_{\text{perihelion}} = m \cdot v_{\text{aphelion}} \cdot r_{\text{aphelion}}
Since the mass m m cancels out, we have:
vperihelion⋅rperihelion=vaphelion⋅raphelion v_{\text{perihelion}} \cdot r_{\text{perihelion}} = v_{\text{aphelion}} \cdot r_{\text{aphelion}}
Substitute the known values:
54 km/s⋅59 A.U.=vaphelion⋅35 A.U. 54 \, \text{km/s} \cdot 59 \, \text{A.U.} = v_{\text{aphelion}} \cdot 35 \, \text{A.U.}

STEP 5

Solve for the speed at aphelion vaphelion v_{\text{aphelion}} :
vaphelion=54 km/s⋅59 A.U.35 A.U. v_{\text{aphelion}} = \frac{54 \, \text{km/s} \cdot 59 \, \text{A.U.}}{35 \, \text{A.U.}}
Calculate the value:
vaphelion=54⋅5935 km/s v_{\text{aphelion}} = \frac{54 \cdot 59}{35} \, \text{km/s}
vaphelion=318635 km/s v_{\text{aphelion}} = \frac{3186}{35} \, \text{km/s}
vaphelion≈91.03 km/s v_{\text{aphelion}} \approx 91.03 \, \text{km/s}
The speed of Halley's comet when it is furthest from the Earth is approximately:
91.03 km/s \boxed{91.03 \, \text{km/s}}

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