9. Let X denote the horizontal distance traveled when a projectile (launched at an angle θ0 with 30° < θ0 < 60°) has attained its maximum height. If the effect of air resistance is taken into account, then which one of the following relations is correct concerning the projectile’s range R?
A. R/2 < X B. (R cos θ0)/2 > X C. (R sin θ0)/2 > X D. R/2 > X
Passage 8 (Questions 1-10)
The gravitational attraction between the Earth (mass M) and an object of mass m on its surface is given approximately by the formula Fgrav = GMm/R2, where G is the universal gravitational constant and R is the radius of the Earth. The formula is exact for a uniform perfect sphere. The gravitational acceleration of an object near the surface of a planet is given by g = GM/R2, which is very nearly 10 m/s2 for the Earth.
The above equation for Fgrav may be generalized so that it is valid both near and far from the surface: Fgrav = GMm/r2 where r is now the distance between the object of mass m and the center of the Earth. On the surface, of course, r = R, so the two formulations are consistent.
The potential energy of an object near the surface of the Earth at a height h above the ground is Ep = mgh. The change in potential energy when an object is moved from height h1 to h2 is
∆Ep = mg(h2 – h1). If an object is falling in a vacuum, energy is conserved so ∆Ep + ∆Ek = 0, where ∆Ek is the change in kinetic energy of the object.
If h is larger than about 0.01R, then Ep = mgh is no longer a good approximation since the gravitational acceleration g decreases as one moves away from the Earth’s surface. To handle problems involving objects far from the surface of the Earth, we use, in analogy with electrostatics, the gravitational potential V = –GM/r where r is the distance of the object from the center of the planet. If an object of mass m is moved from a point at r1 to the velocity of the object. The computed velocities will have the same magnitude; one will be an impact velocity and the other will be an escape velocity.
All of the above can be applied to any celestial body as long as one can ignore air resistance. Some convenient approximations for the relevant constants are G = 6.7 × 10–11 N·m2/kg2, M = 6
× 1024 kg, and R = 6 × 106 m. The Earth is roughly 100 times more massive than the Moon, and the Earth’s radius is about 4 times greater. The Earth–Moon distance is about 60R, and the Earth–Sun distance is about 25,000R. The Sun’s mass is about 300,000M.
1. What would be the acceleration of an object dropped near the surface of the Moon?
A. 0.1 m/s2 weight at this altitude.
A. 1.67 N B. 3.75 N C. 5.00 N D. 7.50 N
3. Assuming the Moon’s orbit is a perfect circle, at what speed does it orbit the Earth?
A. 101 m/s B. 103 m/s C. 105 m/s D. 107 m/s
4. Which one of the following expressions correctly gives the minimum speed with which one must launch a rock of mass m in order to ensure that it escapes the Earth’s gravitational field (air resistance neglected)?
A. 2GM R/ B. 2GMm R/ C. GM/ 2R D. GMm/ 2R
5. If the gravitational force exerted by the Earth on the Moon is denoted F1, and the gravitational force exerted by the Moon on the Earth is F2, then:
A. F1 = 1600F2 B. F1 = 100F2 C. F1 = 6.25F2 D. F1 = F2
6. If α is the ratio of the value of the gravitational constant G on the Earth to its value on the Moon, then which of the following is correct?
A. α ≈ 0.16 B. α ≈ 1 C. α ≈ 6 D. α ≈ 25
7. If two objects of masses m1 and m2 (with m1 > m2) are dropped from a great height above the surface of the Moon, what is the ratio v1/v2 of their impact velocities?
A. 1 magnitude of the change in potential energy?
A. 0 J B. 50 J C. 150 J
D. Cannot be determined from the information given
9. Suppose the Moon suddenly stopped in its orbit and began to fall toward the Earth. Approximate its initial acceleration in terms of g, where g ≈ 10 m/s2. and the Sun is 30 million times more massive than the Moon. Which one of the following best approximates the ratio FSun/FMoon of the gravitational force exerted by the Sun on the Earth to the gravitational force exerted by the Moon on the Earth?
A. 7.5 × 104 B. 2.0 × 102 C. 5.0 × 10–3 D. 1.3 × 10–5
Passage 9 (Questions 1-8)
Earth’s atmosphere currently consists of 78% N2, 21% O2, 1% Ar, and minute quantities of carbon dioxide, hydrogen, neon, helium, krypton, and xenon. This does not include water vapor.
Molecules of atmospheric gas move in random directions with a broad distribution of speeds.
For each gas in the atmosphere, a curve known as the Maxwell–Boltzmann speed distribution may be determined. A typical example of such a curve is shown below. The probability that a molecule has a speed between v1 and v2 is equal to the area below the curve between v1 and v2, as illustrated in Figure 1. Each gas has an associated curve whose details depend on the mass of the molecules and the temperature of the gas. For example, the most probable speed for a gas of with molecular mass m at temperature T (in kelvins, K) is
v0= 2kT m/
where k = 1.38 × 10–23 J/K is Boltzmann’s constant. The distribution of speeds is slightly asymmetrical, favoring higher speeds. Some molecules have speeds much higher than the most probable speed.
v1 v3 speed v
probability density
most probable speed v2
Figure 1 Maxwell–Boltzmann speed distribution The probability of finding a molecule of a gas at temperature T with a speed v within some given range v1 < v < v2 is equal to the area under the curve between v1 and v2.
If a molecule of gas in the atmosphere has a velocity greater than escape velocity, then it will leave the Earth forever as long as it does not collide with any other molecules on its way into outer space. The escape velocity ve for an object close to the surface of the Earth is given by
ve= 2gRE
where RE is the radius of the Earth, and g is the gravitational acceleration.
1. The most probable speed of a helium atom at 300 K is 1.12 km/sec. What is its most probable speed at 600 K?
A. 0.56 km/sec B. 1.58 km/sec C. 2.24 km/sec D. 3.36 km/sec
2. Let the most probable speeds of H2, N2, and O2 at 300 K be denoted vH, vN, and vO, respectively. Which of the following is correctly lists these speeds in order from lowest to highest?
A. vH, vN, vO B. vO, vN, vH C. vH, vO, vN D. vN, vO, vH
3. Precise measurements taken over a period of many years show that the concentration of a Gas X in the atmosphere is constant. Which one of the following best explains this finding?