NDA (II) 2017 GAT Q72: In a vacuum, a five-rupee coin, a feather of a sparrow bird

Q: What is the answer to NDA (II) 2017 GAT Q72?

D. Correct — D, t₁ = t₂ = t₃. The single word that decides this question is 'vacuum'. With no air there is no drag and no buoyancy, so the only force on each object is its weight, and Newton's second law gives an acceleration equal to the weight divided by the mass — which is g for every object, whatever its mass or shape. Falling from rest through the same height h, each takes a time equal to the square root of twice h divided by g, an expression in which neither the mass nor the size appears. The coin, the feather and the mango therefore land together. In ordinary air the feather would drift down long after the other two, because air resistance is large compared with its tiny weight; take the air away and the difference vanishes.

NDA (II) 2017 General Ability Test · Question 72 of 150

Question text verified against the official NDA question paper.

In a vacuum, a five-rupee coin, a feather of a sparrow bird and a mango are dropped simultaneously from the same height. The time taken by them to reach the bottom is t₁, t₂ and t₃ respectively. In this situation, we will observe that

(a)t₁ > t₂ > t₃
(b)t₁ > t₃ > t₂
(c)t₃ > t₁ > t₂
(d)t₁ = t₂ = t₃

Answer

Why

Correct — D, t₁ = t₂ = t₃. The single word that decides this question is 'vacuum'. With no air there is no drag and no buoyancy, so the only force on each object is its weight, and Newton's second law gives an acceleration equal to the weight divided by the mass — which is g for every object, whatever its mass or shape. Falling from rest through the same height h, each takes a time equal to the square root of twice h divided by g, an expression in which neither the mass nor the size appears. The coin, the feather and the mango therefore land together. In ordinary air the feather would drift down long after the other two, because air resistance is large compared with its tiny weight; take the air away and the difference vanishes.

Why the others are wrong

(a)t₁ > t₂ > t₃ — This orders the times by something other than gravity, and it would need the feather to fall faster than the coin, which is not what happens even in air. Nothing in a vacuum can make the times differ at all.
(b)t₁ > t₃ > t₂ — This makes the feather the quickest and the coin the slowest — the reverse of the everyday intuition, and equally wrong. In a vacuum the ordering question does not arise, since the three times are the same.
(c)t₃ > t₁ > t₂ — This is the answer a student gives who has forgotten the word 'vacuum' and is ranking the objects by air resistance. Even in air it gets the order wrong, but the real error is applying an air-resistance argument where there is no air.

Concept

The claim that all bodies fall at the same rate in the absence of air goes back to Galileo and is a direct consequence of the equality of gravitational and inertial mass. Gravity pulls harder on a heavier object, but a heavier object is correspondingly harder to accelerate, and the two effects cancel exactly. The result is that free fall is one of the few motions in which the properties of the falling body simply do not enter.

Scan any falling-body question for the words 'vacuum', 'air resistance neglected' or 'in air', because they change the answer completely. In real air, the shape and density of the body matter enormously — a body reaches terminal velocity when drag has grown to balance weight, and for a feather that happens almost at once and at a very low speed. The classic demonstration is the coin and feather in an evacuated glass tube, and the most famous version was performed on the Moon in 1971, where astronaut David Scott dropped a hammer and a falcon feather together and they struck the surface at the same moment.

Key facts

In a vacuum every body falls with the same acceleration g, regardless of mass, size or shape.
The time to fall from rest through a height h is the square root of 2h/g, with no dependence on mass.
All bodies fall alike because gravitational mass and inertial mass are equal.
In air, drag makes light and broad objects fall more slowly and brings them to a terminal velocity.
The Apollo 15 hammer and feather demonstration on the Moon showed the effect in a natural vacuum.

The same three objects, with air and without

In a vacuum→Only weight acts, the acceleration is g for all three, and t₁ = t₂ = t₃ — the case in the question

In air, the coin→Dense and compact, so drag is small compared with its weight and it falls almost as in a vacuum

In air, the mango→Heavier still, so drag matters even less over a short drop

In air, the feather→Very light with a large surface, so drag balances its weight almost at once and it drifts down slowly

Take the air away and every difference between the three disappears.

Study next

Equations of motion for uniformly accelerated bodiesAcceleration due to gravity and how it varies with altitude, depth and latitudeAir resistance, drag and terminal velocityWeightlessness and free fall inside an orbiting satelliteGalileo's experiments and the equivalence of gravitational and inertial mass

Common traps

Reading past the word 'vacuum' and answering with everyday experience of a falling feather.
Thinking heavier bodies fall faster, even in a vacuum.
Assuming size or shape can matter when there is no medium to push against.

NDA hides a one-word condition such as 'in a vacuum' in the stem and offers three orderings and one equality, so the first job is to find that word.

Related PYQs

NDA_GAT_2016_II_Q782016Open & attempt →

The free fall acceleration g increases as one proceeds, at sea level, from the equator toward either pole. The reason is

(a) Earth is a sphere with same density everywhere
(b) Earth is a sphere with different density at the polar regions than in the equatorial regions
(c) Earth is approximately an ellipsoid having its equatorial radius greater than its polar radius by 21 km
(d) Earth is approximately an ellipsoid having its equatorial radius smaller than its polar radius by 21 km

Answer(c) Earth is approximately an ellipsoid having its equatorial radius greater than its polar radius by 21 km

Sharpens the same quantity from the other side — g is identical for all bodies at a given place, which is this question, but it is not identical everywhere on Earth.

NDA_GAT_2022_I_Q822022Open & attempt →

The energy possessed by a body due to its change in position or shape is called

(a) thermal energy
(b) potential energy
(c) kinetic energy
(d) electric energy

Answer(b) potential energy

Supplies the energy account of the same drop: each of the three objects converts its own potential energy into kinetic energy, and the conversion happens over the same time for all of them.

Practice

practice — not a real PYQ
A body dropped from rest in a vacuum falls through a height h in time t. If a body of twice the mass is dropped from the same height in the same vacuum, the time taken is
- (a)t
- (b)2t
- (c)t divided by 2
- (d)4t
Answer(a) t — the time of free fall does not depend on the mass, since every body has the same acceleration g.
practice — not a real PYQ
A body falling through air reaches terminal velocity when
- (a)the drag force becomes equal to its weight
- (b)its acceleration becomes equal to g
- (c)its kinetic energy becomes zero
- (d)the air pressure around it becomes zero
Answer(a) the drag force becomes equal to its weight — the net force is then zero and the body falls at a steady speed.