Questions from Kinematics (Including Karnataka CET 2008 Question)

“God used beautiful mathematics in creating the world.”

– P.A.M. Dirac

Today we shall discuss a few questions (MCQ) in the section, ‘kinematics in one dimension’. The questions I give you are meant for testing your knowledge, comprehension and the ability for applying what you have learned in this section.

(1) The velocity-time graphs of a car and a motor bike traveling along a straight road are shown in the adjoining figure. At time t = 0 they have the same position co-ordinate.  Pick out the correct statement from the following:

(a) At time t = 0 the car and the motor bike are at rest.

(b) At time t = 0 the car is at rest but the motor bike is moving.

(c) The distances traveled by the car and the motor bike in time t₁ are equal.

(d) The distance traveled by the bike in time t₁ is twice the distance traveled by the car in the same time.

(e) The distance traveled by the bike in time t₁ is half the distance traveled by the car in the same time.

You can easily conclude that options (a) and (b) are wrong.

To check the remaining options we use the equation of linear motion, s = ut + ½ at² where s is the displacement in time t, u is the initial velocity and a is the uniform acceleration.

The distance traveled by the car in time t₁ is v_ct₁ where v_c (let us say) is the constant velocity of the car.

[The velocity of the car is constant since the velocity-time graph of the car is parallel to the time axis].

The motion of the motor bike is uniformly accelerated and the acceleration a is given by

            a =  v_c/t₁

[Note that the velocity of the motor bike changes from 0 to v_c in time t₁.

The distance s traveled by the motor bike in time t₁ is given by

            s = ut + ½ at² = 0 + (½)×(v_c/t₁) t₁²

Or, s = v_ct₁/2

Therefore the distance traveled by the bike in time t₁ is half the distance traveled by the car in the same time [Option (e)].

(2) A student standing at the edge of a cliff throws a stone of mass m vertically upwards with speed v. It strikes the ground at the foot of the cliff with speed v₁. The student then throws another stone of mass m/4 vertically downwards with speed v. It strikes the ground at the foot of the cliff with speed v₂. If air resistance is negligible, v₁ and v₂ are related as

(a) v₁ = v₂

(b) v₁ = v₂/2

(c) v₁ = 2v₂

(d) v₁ = v₂/4

(e) v₁ = 4v₂

While returning, the sphere of mass m has downward speed v when it passes the edge of the cliff. For calculating the speed with which it strikes the ground at the foot of the cliff, the situation is similar to that of the stone of mass m/4 thrown downwards. Obviously both stones will strike the ground with the same speed [Option (a)].  

(3) A body is projected vertically upwards. The times corresponding to height h while ascending and descending are t₁ and t₂ are respectively. Then the velocity of projection is (g is acceleration due to gravity)

(1) g√(t₁t₂)

(2) gt₁t₂/(t₁+t₂)

(3) g√(t₁t₂)/2

(4) g(t₁+t₂)/2

The above question appeared in Karnataka CET 2008 question paper.

We have the following two equations to give h:

            h = ut₁ –  (½) g t₁²…………….(i)

            h = ut₂ –  (½) g t₂²…………….(ii)

(We have taken the displacement h and the initial velocity u (both upwards) as positive and that’s why the acceleration due to gravity g is negative).

(ii) – (i) gives u(t₂ – t₁) = (½) g(t₂² – t₁²)

Therefore u = (½) g(t₂² – t₁²)/(t₂ – t₁) = g(t₁+t₂)/2

You can find a few more multiple choice practice questions (with solution) in this section here.

Multiple Choice Questions on Dimensions of Physical Quantities [Including KEAM (Engineering) 2011 question]

I believe in standardizing automobiles, not human beings

– Albert Einstein

In most of the entrance examination question papers you will find at least one question on dimensions of physical quantities. Here are a few typical multiple choice questions in this section:

(1) Which one among the following quantities is a dimensional constant?

(a) Dielectric constant of water

(b) Speed of light in free space

(c) Viscosity of water

(d) Ratio of specific heats of a diatomic gas

(e) Reynolds number

Options (a) and (c) are not constants since the dielectric constant and viscosity depend on other parameters. Options (d) and (e) are dimensionless numbers.

[Reynolds number (used in assessing the turbulence of fluids) is the ratio of inertial force to force of viscosity]

The correct option is the speed of light in free space which is a fundamental constant with dimensions LT^–1.

(2) If F denotes force and t time, then in the equation F = at^–1 + bt², the dimensions of a and b respectively are

(a) LT^–4 and LT^–1

(b) LT^–1 and LT^–4

(c) MLT^–4 and MLT^–1

(d) MLT^–1 and MLT^–4

(e) MLT^–3 and MLT^–2

The above question appeared in Kerala Engineering Entrance ((KEAM) 2011 question paper.

The dimensions of at^–1 and bt² have to be that of force which is MLT^–2. Therefore the dimensions of a must be MLT^–1 and the dimensions of b must be MLT^–4.

The correct option is (d).

The following question appeared in Karnataka CET 2004 question paper:

(3) The physical quantity having the same dimensions as Planck’s constant h is

(1) Boltzmann constant

(2) force

(3) linear momentum

(4) angular momentum

Most of you will remember that the unit of h is joule second. Therefore h has the dimensions of the product of work and time which is ML²T^–2×T = ML²T^–1.

Angular momentum is the moment of linear momentum and has dimensions L×MLT^–1 = ML²T^–1.

The correct option is (4).

The following question appeared in EAMCET 2008 Engineering Entrance Exam question paper. You will answer it in no time if you remember that the dimensions of Planck’s constant are those of angular momentum.

(4) The energy (E), angular momentum (L) and universal gravitational constant (G) are chosen as fundamental quantities. The dimensions of universal gravitational constant in the dimensional formula for Planck’s constant (h) is

(a) zero

(b) – 1

(c) 5/3

(d) 1

Since the dimensions of Planck’s constant are those of angular momentum, it follows that the dimensions of universal gravitational constant in the dimensional formula for Planck’s constant (h) is zero.

[In terms of E, L and G which are assumed as fundamental quantities in the above question, Planck’s constant (h) has zero dimension in E, one dimension in L and zero dimension in G].