physics : class11 Radial and Transverse Velocity : Second Law of Kepler : Gravitation MCQ Questions and Answers
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Radial and Transverse Velocity
MCQ questions and answers.
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MCQ Questions
1. A planet moves in an orbit under a central gravitational force. At one point its distance from the Sun is 2.0 × 10^10 m and its transverse velocity is 6.0 km/s. At another point its distance is 5.0 × 10^10 m. What is its transverse velocity at the second point?
A.
1.2 km/s
B.
2.4 km/s
C.
6.0 km/s
D.
15.0 km/s
Correct Answer:
2.4 km/s
Explanation:
Kepler's second law gives constant areal velocity: (1/2)r v_t = constant. Hence r_1v_{t1}=r_2v_{t2}, so v_{t2}=(2.0/5.0)×6.0=2.4 km/s.
2. At a particular point in an elliptical orbit, a planet has radial velocity 8 m/s and transverse velocity 15 m/s. What is the angle made by its velocity vector with the outward radial direction?
A.
28.1°
B.
36.9°
C.
53.1°
D.
61.9°
Correct Answer:
61.9°
Explanation:
For the angle α measured from the outward radial direction, tan α = v_t/v_r = 15/8. Therefore α = tan⁻¹(15/8) ≈ 61.9°.
3. A planet is at a point where its distance from the Sun is 3.0 × 10^10 m and its radial velocity is −4.0 km/s. If its transverse velocity is 12 km/s, what is the magnitude of its total velocity?
A.
8.0 km/s
B.
12.6 km/s
C.
16.0 km/s
D.
20.0 km/s
Correct Answer:
12.6 km/s
Explanation:
Radial and transverse velocities are perpendicular components. Thus v = √(v_r²+v_t²) = √(4²+12²) = √160 ≈ 12.6 km/s.
4. A planet sweeps out area at the constant rate 3.0 × 10^15 m²/s. When its distance from the Sun is 2.0 × 10^10 m, what is its transverse velocity?
A.
75 m/s
B.
150 m/s
C.
300 m/s
D.
600 m/s
Correct Answer:
300 m/s
Explanation:
The areal velocity is dA/dt = (1/2)rv_t. Therefore v_t = 2(dA/dt)/r = 2(3.0×10^15)/(2.0×10^10) = 300 m/s.
5. At the nearest point of a bound planetary orbit around the Sun, which statement about the planet's radial and transverse velocities is correct?
A.
Radial velocity is maximum and transverse velocity is zero
B.
Radial velocity is zero and transverse velocity is maximum
C.
Both radial and transverse velocities are zero
D.
Radial velocity and transverse velocity have equal nonzero magnitudes
Correct Answer:
Radial velocity is zero and transverse velocity is maximum
Explanation:
At the nearest point, the radial distance is momentarily not changing, so v_r = dr/dt = 0. Since rv_t is constant and r is minimum, the transverse velocity is maximum.
6. A planet has transverse velocity 20 km/s at a distance r from the Sun. At another point its transverse velocity is 8 km/s. If its radial distances at the two points are r and 2.5r respectively, what can be concluded?
A.
The data obey Kepler's second law exactly
B.
The second distance must be 1.6r
C.
The second transverse velocity should be 12.5 km/s
D.
The planet must have zero radial velocity at both points
Correct Answer:
The data obey Kepler's second law exactly
Explanation:
Kepler's second law requires r_1v_{t1}=r_2v_{t2}. Here r×20 = (2.5r)×8 = 20r, so the data satisfy constant areal velocity.
7. The velocity of a planet at a certain point makes an angle of 120° with the outward radial direction, and its speed is 10 km/s. What are the signs of its radial and transverse velocity components?
A.
v_r positive and v_t positive
B.
v_r positive and v_t negative
C.
v_r negative and v_t positive
D.
v_r negative and v_t negative
Correct Answer:
v_r negative and v_t positive
Explanation:
The radial component is v_r = v cos120°, which is negative, indicating inward motion. Taking the direction of increasing angular coordinate as positive, the transverse component is positive for the usual prograde orbital motion.
8. A planet moves with constant areal velocity. At two positions, its radial velocities are 6 m/s and −6 m/s, respectively, while its transverse velocities are 10 m/s and 15 m/s. What is the ratio of its distances from the Sun at these positions, r_1:r_2?
A.
2:3
B.
3:2
C.
4:9
D.
9:4
Correct Answer:
2:3
Explanation:
The radial component does not enter the areal-velocity relation. Since r_1v_{t1}=r_2v_{t2}, r_1:r_2=v_{t2}:v_{t1}=15:10=3:2. Wait, writing the ratio directly gives r_1/r_2=15/10=3/2, so the correct option is 3:2.
9. For a planet in a central gravitational field, its position vector is perpendicular to its velocity vector at an instant. Which statement follows at that instant?
A.
The radial velocity is equal to the total velocity
B.
The transverse velocity is zero
C.
The radial velocity is zero and the total velocity equals the transverse velocity
D.
The radial and transverse velocities have equal magnitudes
Correct Answer:
The radial velocity is zero and the total velocity equals the transverse velocity
Explanation:
The velocity is perpendicular to the position vector only when it has no radial component. Thus v_r=0 and the entire velocity is transverse, so v=v_t.
10. A planet's transverse velocity changes from 4 km/s to 16 km/s as it moves along its orbit. If its initial radial distance is 8.0 × 10^10 m, what is its radial distance at the later point, assuming Kepler's second law?