Answer :
Final answer:
The speed at aphelion for comet TOTAS is approximately 10.8 km/s, calculated using the conservation of angular momentum which allows for the simplification and cancellation of common terms.
Explanation:
The speed at aphelion for a comet orbiting the Sun can be determined using the conservation of angular momentum, which states that the angular momentum at perihelion is equal to the angular momentum at aphelion. The formula to use is Lperihelion = m * vperihelion * rperihelion = Laphelion = m * vaphelion * raphelion, where m is the mass of the comet, v is the velocity, and r is the distance from the Sun. Given the provided data: the perihelion is 1.69 AU, the aphelion is 4.40 AU, and the speed at perihelion is 28 km/s, we can use the formula to solve for the speed at aphelion.
First, convert the units for the perihelion and aphelion distances into meters by multiplying with the conversion factor (1 AU = 1.496 × 1011 m). Then, apply the formula:
Calculate the angular momentum at perihelion: Lperihelion = m * 28 km/s * 1.69 AU * (1.496 × 1011 m/AU).
Using the conservation of angular momentum, set Lperihelion equal to Laphelion to solve for vaphelion:
vaphelion = Lperihelion / (m * 4.40 AU * (1.496 × 1011 m/AU)).
Cancel out the comet's mass since it is common to both sides of the equation.
After simplifying, solve for the speed at aphelion.
Since the mass of the comet cancels out, and the AU to meters conversion is the same for both perihelion and aphelion, we are left with the simplified relationship:
vaphelion = (28 km/s * 1.69 AU) / 4.40 AU
Inserting the values:
vaphelion ≈ (28 km/s * 1.69) / 4.40 ≈ 10.8 km/s
Therefore, the speed of comet TOTAS at its aphelion is approximately 10.8 km/s.
Answer:
The speed at the aphelion is 10.75 km/s.
Explanation:
The angular momentum is defined as:
[tex]L = mrv[/tex] (1)
Since there is no torque acting on the system, it can be expressed in the following way:
[tex]t = \frac{\Delta L}{\Delta t}[/tex]
[tex]t \Delta t = \Delta L[/tex]
[tex]\Delta L = 0[/tex]
[tex]L_{a} - L_{p} = 0[/tex]
[tex]L_{a} = L_{p}[/tex] (2)
Replacing equation 1 in equation 2 it is gotten:
[tex]mr_{a}v_{a} =mr_{p}v_{p}[/tex] (3)
Where m is the mass of the comet, [tex]r_{a}[/tex] is the orbital radius at the aphelion, [tex]v_{a}[/tex] is the speed at the aphelion, [tex]r_{p}[/tex] is the orbital radius at the perihelion and [tex]v_{p}[/tex] is the speed at the perihelion.
From equation 3 v_{a} will be isolated:
[tex]v_{a} = \frac{mr_{p}v_{p}}{mr_{a}}[/tex]
[tex]v_{a} = \frac{r_{p}v_{p}}{r_{a}}[/tex] (4)
Before replacing all the values in equation 4 it is necessary to express the orbital radius for the perihelion and the aphelion from AU (astronomical units) to meters, and then from meters to kilometers:
[tex]r_{p} = 1.69 AU x \frac{1.496x10^{11} m}{1 AU}[/tex] ⇒ [tex]2.528x10^{11} m[/tex]
[tex]r_{p} = 2.528x10^{11} m x \frac{1km}{1000m}[/tex] ⇒ [tex]252800000 km[/tex]
[tex]r_{a} = 4.40 AU x \frac{1.496x10^{11} m}{1 AU}[/tex] ⇒ [tex]6.582x10^{11} m[/tex]
[tex]r_{p} = 6.582x10^{11} m x \frac{1km}{1000m}[/tex] ⇒ [tex]658200000 km[/tex]
Then, finally equation 4 can be used:
[tex]v_{a} = \frac{(252800000 km)(28 km/s)}{(658200000 km)}[/tex]
[tex]v_{a} = 10.75 km/s[/tex]
Hence, the speed at the aphelion is 10.75 km/s.
