At perihelion on Feb. 9, 1986, Halley’s comet was 8.79 × 107 km from the sun and was
moving at a speed of 54.6 km s−1 relative to the sun. Calculate its speed (a) when the comet
was 1.16 × 108 km from the sun, and (b) at its next aphelion in the year 2024, when the
comet will be 5.28 × 109 km from the sun.
Well, calculating speeds in space can be tricky, but let's give it a shot!
(a) To calculate the speed when the comet was 1.16 × 10^8 km from the sun, we can use the conservation of angular momentum. Since the comet's angular momentum is conserved, we can say that the product of its initial distance from the sun and its initial speed should be equal to the product of its final distance and its final speed.
So, we have (8.79 × 10^7 km) * (54.6 km/s) = (1.16 × 10^8 km) * (V), where V is the speed we want to calculate.
Solving for V, we get V = (8.79 × 10^7 km) * (54.6 km/s) / (1.16 × 10^8 km) ≈ 41.46 km/s.
(b) Now, let's calculate the speed at the next aphelion in the year 2024, when the comet will be 5.28 × 10^9 km from the sun. Applying the same conservation of angular momentum principle, we have:
(8.79 × 10^7 km) * (54.6 km/s) = (5.28 × 10^9 km) * (V'), where V' is the speed we want to calculate.
Solving for V', we get V' = (8.79 × 10^7 km) * (54.6 km/s) / (5.28 × 10^9 km) ≈ 0.906 km/s.
So, the comet's speed at its next aphelion in 2024 will be approximately 0.906 km/s.
Keep in mind that these calculations are based on the simplifying assumption of conservation of angular momentum, so the results may not be 100% accurate. But hey, at least we gave it a good try!
To calculate the speed of Halley's comet at different distances, we can use the principle of conservation of angular momentum. Angular momentum is given by the equation:
L = r * v * m,
where L is the angular momentum, r is the distance from the sun, v is the speed, and m is the mass of the comet (which we assume is constant).
1. To calculate the speed when the comet was 1.16 × 10^8 km from the sun:
First, we need to find the angular momentum at perihelion (r1 = 8.79 × 10^7 km) using the given parameters:
L1 = r1 * v1 * m,
Next, we can use the conservation of angular momentum to find the speed at the new distance (r2 = 1.16 × 10^8 km):
L1 = L2,
r1 * v1 * m = r2 * v2 * m,
v2 = (r1 * v1) / r2.
Substituting the given values, we have:
v2 = (8.79 × 10^7 km * 54.6 km/s) / (1.16 × 10^8 km) = 4.14 km/s.
Therefore, when the comet is 1.16 × 10^8 km from the sun, its speed is approximately 4.14 km/s.
2. To calculate the speed at the next aphelion in the year 2024 (r3 = 5.28 × 10^9 km):
Similarly, we can use the conservation of angular momentum to find the speed at the new distance (r3 = 5.28 × 10^9 km) using the previous speed (v2):
v3 = (r2 * v2) / r3,
v3 = (1.16 × 10^8 km * 4.14 km/s) / (5.28 × 10^9 km) = 0.0091 km/s.
Therefore, at its next aphelion in 2024, Halley's comet will be moving at a speed of approximately 0.0091 km/s relative to the sun.
To solve this problem, we can use the principle of conservation of angular momentum. According to this principle, the angular momentum of an object revolving around a fixed point remains constant as long as no external torque acts on it.
The formula for angular momentum is given by:
L = mvr
Where:
L = angular momentum
m = mass of the object
v = velocity of the object
r = distance from the object to the fixed point (in this case, the sun)
Now, let's calculate the angular momentum at perihelion (r1 = 8.79 × 10^7 km) using the given speed (v1 = 54.6 km/s):
L1 = m * v1 * r1
Next, we can use the conservation of angular momentum to calculate the speed at a different distance, r2. We'll use the same angular momentum (L1) and solve for v2:
L1 = m * v2 * r2
Now, let's plug in the values and solve for v2:
m * v1 * r1 = m * v2 * r2
The mass of the comet cancels out:
v2 = (v1 * r1) / r2
(a) Given that r2 = 1.16 × 10^8 km, we can calculate v2:
v2 = (54.6 km/s * 8.79 × 10^7 km) / (1.16 × 10^8 km) = 41.4 km/s
Therefore, the speed of the comet when it is 1.16 × 10^8 km from the sun is 41.4 km/s.
(b) For the next aphelion, r2 = 5.28 × 10^9 km. Let's calculate v2:
v2 = (54.6 km/s * 8.79 × 10^7 km) / (5.28 × 10^9 km) = 0.914 km/s
Therefore, the speed of the comet at its next aphelion (in 2024) when it is 5.28 × 10^9 km from the sun is 0.914 km/s.