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Quantitative Reasoning – How fast will you fall down when released from 100 metres above the moon’s surface!

PSA 2014_XI_Q22. Questions (i-iv) : Read the following passage and answer the questions that follow :

Everyone knows the expression what goes up,must come down but have you ever wondered exactly why things must come down ? The fact is that all objects on and near the Earth are pulled towards the Earth’s centre by the Earth’s gravity. The force of gravity makes all falling objects travel faster and faster the longer they are falling. This is called gravitational acceleration. On Earth, the speed of a dropped object progressively increases by about 9.8 m/s for every second that it falls. This is just like what happens to a car’s speed as it accelerates away from a stop sign. A dropped hammer will have a speed of 9.8 m/s after its first second of travel, a speed of 19.6 m/s after two seconds and so
on.
Other large masses such as the Moon, the Sun and other planets also have gravity, although it may be stronger or weaker than on Earth. The gravity that an object experiences on a planet’s surface is directly proportional to the planet’s mass and inversely proportional to the planet’s radius squared. So, a planet that has the same radius as the Earth (6400 km) but has twice the mass will have gravity that is twice as strong as the Earth’s. A planet that has the same mass as the Earth (6 × 1024 kg) but twice the radius will have gravity that is four times weaker than Earth’s.

The Moon’s gravity, for example, is one-sixth that of the Earth’s. A hammer dropped on the Moon will fall much more slowly there than it would on the Earth — it would increase in speed by only 1.6 m/s every second that it fell. On Jupiter, where gravity is two and a half times greater than on Earth, the hammer would fall more quickly, increasing its speed by 24.5 m/s every second it fell.

(i) The planets Mercury, Mars and Venus have masses that have values quite close to the value of the mass of the earth. The masses of Uranus and Neptune are both nearly 20 times the mass of the earth, with Uranus heavier of the two. Saturn has a mass close to 100 times that of the earth while the mass of Jupiter, the heaviest of all planets, is nearly 320 times that of the earth.
The radii of the planets, in terms of the radius of the earth vary in the following sequence :
The radii of Mercury as well as Mars are nearly 40% of that of the earth with Mars bigger of the two.
The radius of Venus, is close to but slightly smaller than that of the earth.
Neptune’s radius is nearly 4 times that of the earth while that of Uranus is nearly 3.9 times that of the earth.
Saturn has a radius nearly 9.5 times that of the earth while Jupiter, the largest of all planets, has a radius 11.2 times that of the earth.
These data values suggest that the planet, having a gravity half that of Jupiter, is likely to be :
(1) Mercury

(2) Earth

(3) Saturn

(4) Neptune

(ii)Mars has one-tenth of the Earth’s mass and a radius that is one-half of the Earth’s radius. Therefore, the value of Mars gravity divided by Earth gravity will be
(1) 2.5

(2) 0.2

(3) 0.4

(4) 5.0

(iii)A rock is dropped on a planet that has the same mass as the Earth. Its speed increases by 1.09 m/s every second that it falls. Which of the following statements is true ?
(1) The planet has a radius 9 times smaller than the Earth’s radius.
(2) The planet’s radius is 3 times larger than the Earth’s radius.
(3) The planet’s radius is 3 times smaller than the Earth’s radius.
(4) The planet has a radius 9 times larger than the Earth’s radius.

(iv) In one experiment, the following results were obtained :
When a hammer was dropped from a height on the earth, its speed at the end of one second was 9.8 m/s. The values of these speeds at the end of two seconds, three seconds, four seconds and five seconds of the hammer’s fall were measured to be 19.6 m/s, 29.4 m/s, 39.2 m/s and 49.0 m/s, respectively.
When an identical hammer was dropped from a height on the moon, its speeds at the end of one second, two seconds, three seconds, four seconds and five seconds of its fall were measured to be 1.6 m/s, 3.2 m/s, 4.8 m/s, 6.4 m/s and 8.0 m/s respectively.
In a second experiment, two identical hammers were dropped on the earth, and on the moon, from the same height at exactly the same time. It was observed that the earth hammer was travelling at a speed of 68.6 m/s when it hit the ground.
Which of the following is nearest to the speed of the moon hammer when the earth hammer strikes the ground ?
(1) 12 m/s

(2) 4 m/s

(3) 6 m/s

(4) 11 m/s

Ans: (i) Assuming ‘m’ and ‘r’ to be mass and radius of earth, tabulating the mass and radii of other planets:

PSA 2014_XI_Q21_1

Gravitational force on Jupiter, Gj∝ 320m/(11.2r)²

Gj∝ (320/125.4) m/(r²) or Gj∝ 2.55 m/(r²)

Gravitational force on Mercury, Gm∝ m/(0.4r)² or Gm∝ 2.5 m/(r)²

Gravitational force on Earth, Ge∝ m/(r)²

Gravitational force on Saturn, Gs∝ 100m/(9.5r)²  or Gs∝ 1.11 m/(r)²

Gravitational force on Neptune, Gn∝ 20m/(4r)² or Gn∝ 1.25 m/(r)²

Correction option (4). Neptune. The other close option (3) is bit on the lower side.

(ii) Gravitational force on Mars, Gm∝ 0.1m/(0.5r)², where ‘m’ and ‘r’ to be mass and radius of earth.

Gm∝ 0.4m/(r)². Option (3) 0.4

(iii) Acceleration due to gravity on earth= 9.8 m/s²

Acceleration due to gravity on the other planet= 1.09 m/s²

Gravity of the other planet = 1/9th of that on earth

Since mass is same, radius must be larger than earth. Since, the gravity is inversely proportional to square of the radius, the radius of the other planet must be 3 times the earth’s radius. Option (2).

(iv) After 1 second, the hammer’s speed on earth =9.8 m/s

After 1 second, the hammer’s speed on moon=1.6 m/s

Hammer’s speed on earth is little more than 6 times that on moon. Looking at the options closer values are in (1) and (4).

First option: 12 m/s. Six times this value (72) crosses the speed 68.6 m/s.

Fourth option: 11 m/s. Six times this value (66) is closest to the speed 68.6 m/s. Since, the value has to be little more than 6 times, this is the correct option.

Option 4.

 

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