Explanation

(Option C) Unequal displacements are made in equal intervals of time: This means that in each equal interval of time, the object is covering different distances. For example, in each second, the object travels different distances such as 1 meter in the first second, 2 meters in the second second, etc. This directly indicates a change in speed over time, which is the precise definition of variable velocity. The object’s velocity is not constant; it varies because the distance traveled per unit time changes.

(Option A) Unequal distances are covered in equal intervals of time: This means that in equal intervals of time, the object is covering different distances. For example, in the first second, it might travel 2 meters, in the next second, it might travel 3 meters, and so on. This indicates that the object's speed is changing, which is characteristic of variable velocity.

(Option B) Equal displacements are made in unequal intervals of time: This means that the object travels the same distance in different amounts of time. For instance, it might take 2 seconds to travel 5 meters initially, then 3 seconds to travel the next 5 meters. This suggests that the object's speed is changing, but it's not the standard way to describe variable velocity. Instead, this option implies that the object’s velocity is changing but does not clearly state how it relates to time intervals being equal.

(Option D) Equal displacements are made in equal intervals of time: This means that the object travels the same distance in each equal interval of time. For example, it travels 5 meters every second consistently. This describes a situation where the object's speed is constant, meaning it has a constant or uniform velocity, not a variable one.

Subject: Physics

Topic: Force and Motion

Subtopic: Velocity

Difficulty Level: Moderate

Velocity time graph for variable velocity

The slope of the position versus time graph shown above would equal 20 cm divided by 0.1 sec or 200 cm/sec.

The following graph displays this exact same information in a new format, a velocity versus time graph.

Explanation

(Option C) Change in Momentum: Change in momentum is what results from the product of force and time. This product is known as impulse. The formula for impulse J is:

J=FΔt

Where F is force and Δt is the time duration over which the force acts.

Impulse causes a change in the momentum Δp of an object:

J=Δp=mΔv

Where m is mass and Δv is the change in velocity.

Therefore, the product of force and time equals the change in momentum.

(Option A) Angular Momentum: Angular momentum is a measure of the quantity of rotation of an object and is a product of its moment of inertia and its angular velocity. The formula for angular momentum L is:

L=Iω

Where I is the moment of inertia and ω is the angular velocity.

It is not directly related to the product of force and time.

(Option B) Force: Force is a push or pull acting on an object resulting from the object's interaction with another object. The formula for force F is:

F=ma

Where m is mass and a is acceleration.

The product of force and time gives something else (impulse), not force itself.

(Option D) Velocity: Velocity is the speed of an object in a specific direction. The formula for velocity v is:

v=d/ t

Where d is distance and t is time.

It is not related to the product of force and time directly.

Subject: Physics

Topic: Force and motion

Subtopic: Linear momentum

Difficulty Level: Easy

Linear Momentum

Explanation

(Option A) Reaction force on a body is always balanced by the action force: This statement is incorrect. Newton's Third Law states that for every action, there is an equal and opposite reaction. However, the action and reaction forces act on different bodies, not on the same body. For instance, if you push a wall, the wall pushes back with an equal and opposite force. These forces act on different objects (you and the wall), not on the same object. Therefore, they are not "balanced" on a single body.

(Option B) Reaction and action forces are always equal: This statement is correct. According to Newton's Third Law, action and reaction forces are always equal in magnitude but opposite in direction. If object A exerts a force on object B, then object B exerts an equal and opposite force on object A.

(Option C) Action and reaction forces never act on the same body: This statement is correct. Action and reaction forces always act on different bodies. For example, when a book rests on a table, the book exerts a downward force on the table (action), and the table exerts an upward force on the book (reaction). These forces act on two different objects.

(Option D) Newton’s Third Law is always valid in all situations: This statement is correct. Newton's Third Law holds true in all situations involving forces. There is always an equal and opposite reaction force for every action force, regardless of the scenario.

Subject: Physics

Topic: Force and Motion

Subtopic: Newton's third law of motion

Misleading Option: Option A might be misleading for some students because it involves a common misconception that action and reaction forces balance each other on the same body.

Difficulty Level: Moderate

Explanation

(Option B) 10 m/s:

We'll analyze this problem assuming a simplified scenario where:

Friction, air resistance, and other energy losses are ignored.

The athlete's mass remains constant during the vault.

We focus on the conversion of kinetic energy (KE) to gravitational potential energy (GPE) at the peak of the vault.

Conservation of Mechanical Energy:

We'll use the principle of conservation of mechanical energy, which states that the total mechanical energy of a closed system (no external work done) remains constant. In this case, the closed system is the athlete. Mathematically, this can be expressed as:

KE (initial) + PE (initial) = KE (final) + PE (final)

Since the athlete reaches maximum height at the peak of the vault, their final kinetic energy (KE_final) is negligible (very close to zero). Additionally, the initial potential energy (PE_initial) is zero as the athlete starts from the ground.

Therefore, the equation simplifies to:

KE (initial) = PE (final)

Calculating the Required Speed:

KE (initial): This is the kinetic energy the athlete possesses when they plant the pole. We need to find the speed (v) associated with this energy.

PE (final): This is the gravitational potential energy the athlete has at the peak of the vault, which depends on their mass (m), acceleration due to gravity (g), and the height reached (h = 5 m in this case).

Formulas:

KE (initial) = 1/2 * m * v² (where m is mass and v is speed)

PE (final) = m * g * h (where g is acceleration due to gravity and h is height)

Setting Up the Equation:

Since KE (initial) needs to equal PE (final):

1/2 * m * v² = m * g * h

We can divide both sides by the athlete's mass (m) to cancel it out (assuming it doesn't change during the vault):

1/2 * v² = g * h

Solving for Speed (v):

v² = 2 * g * h

v = sqrt(2 * g * h)

Plugging in the Values:

g (acceleration due to gravity) ≈ 9.8 m/s²

h (height) = 5 m

v = sqrt(2 * 9.8 m/s² * 5 m)

v ≈ 10 m/s

(Option A) 5 m/s: This speed is too low. The athlete wouldn't have enough kinetic energy to reach a height of 5 meters.

(Option C & D) 15 m/s and. 20 m/s: While these speeds would provide enough energy to reach 5 meters (or even higher), they're not the most efficient use of the athlete's energy in this simplified scenario.

Subject: Physics

Topic: Work and Energy

Subtopic: Conversion of Kinetic Energy to Potential Energy

Difficulty Level: Moderate

Pole Vault

Explanation

(Option A) Concave Lens:

Trough of a Wave: The trough is the lowest point in a wave.

Concave Lens: A concave lens is thinner at the center than at the edges. It diverges light rays that pass through it.

Action Similarity: The trough of a wave behaves like a concave lens because it causes light rays passing through it to diverge. This happens due to the curvature of the trough which bends the light away from the central axis.

(Option B) Convex Lens:

Convex Lens: A convex lens is thicker at the center than at the edges. It converges light rays that pass through it.

Why Not This Option: The convex lens focuses light rays to a point, which is the opposite of what the trough of a wave does. Therefore, the trough of a wave does not act like a convex lens.

(Option C) Convex Mirror:

Convex Mirror: A convex mirror curves outward and diverges light rays, making objects appear smaller.

Why Not This Option: Although a convex mirror also diverges light rays, it reflects light rather than allowing it to pass through. A trough of a wave affects light by transmission (like a lens) rather than by reflection, making this option incorrect.

(Option D) Plane Mirror:

Plane Mirror: A plane mirror reflects light without changing its convergence or divergence.

Why Not This Option: A plane mirror does not alter the path of light in the same way a concave lens or the trough of a wave does. It simply reflects the light back without any change in direction apart from the reflection angle. Hence, the trough of a wave does not act like a plane mirror.

 Subject: Physics

Topic: Waves

Subtopic: Trough

Difficulty Level: Moderate

Explanation

Given:

Two positive point charges

Distance between charges: 2 meters

Point of interest: Mid-point between the charges

First, let's understand the electric potential due to a point charge. The electric potential V at a distance r from a point charge Q is given by:

V=kQ/ r

Where k is Coulomb's constant (k≈8.99×10⁹ Nm²/C²).

Now, let's calculate the electric potential at the midpoint.

Since the charges are 2 meters apart, the distance from each charge to the midpoint is:

r=2 / 2=1 meter

If each charge is Q, the potential due to one charge at the midpoint is:

V1=kQ / 1=kQ

Since there are two identical charges, the total potential at the midpoint is:

V_total=V1+V2=kQ+kQ=2kQ

Therefore, the electric potential at the midpoint is indeed double the potential due to a single charge, making Option A: Added to double the correct choice.

(Option A) Added to double: This suggests that the electric potential at the midpoint due to both charges will be twice the potential due to a single charge. Since both charges are positive and the midpoint is equidistant from both charges, the potentials add up constructively.

(Option B) Reduced to half: This would imply that the combined potential at the midpoint is half of what it would be due to a single charge. This is not true because potentials from point charges add algebraically, and there is no mechanism in this scenario that would reduce the potential.

(Option C) Remains the same (no effect): This option suggests that the presence of the second charge does not affect the potential due to the first charge at the midpoint. This is incorrect because potentials due to multiple charges do indeed superimpose.

(Option D) Cancel each other effect: For the potentials to cancel each other out, the charges would need to be of opposite signs. Since both charges are positive, their potentials add up rather than cancel out.

Subject: Physics

Topic: Electrostatic

Subtopic: Electric potential

Difficulty Level: Moderate

Electric Potential Due to Multiple Charges

Explanation

(Option B) Receding the Earth: When an object, like a galaxy, is moving away from the Earth, the light it emits shifts towards the red end of the spectrum. This phenomenon is known as a redshift. The Doppler effect causes the wavelengths to stretch, making them longer as the source moves away from the observer.

Key Concept: Redshift indicates a receding object because the wavelengths are     stretched out.

(Option A) Approaching Earth: When an object, such as a galaxy, is moving towards the Earth, the light it emits shifts towards the blue end of the spectrum. This is known as a blueshift. The reason for this shift is due to the Doppler effect, where the wavelength of light decreases as the source moves closer to the observer.

Key Concept: Blueshift indicates an approaching object because the wavelengths are compressed.

(Option C) Stationary: If the galaxy were stationary relative to the Earth, the wavelengths of light would not shift significantly towards either the red or blue end of the spectrum. The light would appear mostly unchanged in terms of wavelength, remaining at its original position in the spectrum.

Key Concept: No significant shift in wavelength occurs if there is no relative            motion.

(Option D) Approaching Earth or is stationary: This is incorrect because it suggests two conflicting scenarios. As explained above, an approaching galaxy would exhibit a blueshift, while a stationary galaxy would show no shift. The key indicator of movement away from Earth (receding) is the redshift, and not a blueshift or no shift.

Key Concept: Redshift and blueshift are distinct indicators of movement. A stationary object does not exhibit these shifts.

Subject: Physics

Topic: Waves

Subtopic: Doppler effect

Difficulty Level: Moderate

Doppler Effect

Explanation

The resistance R of a wire is given by the formula:

R=ρL/ A

where:

ρ is the resistivity of the material,

L is the length of the wire,

A is the cross-sectional area of the wire.

Let's analyze the given situation step-by-step.

Original Resistance: R1=ρL₁/ A₁

New Dimensions:

The length of the wire becomes twice its original value: L2=2L1 ​

The area becomes half of its original value: A2=(1/ 2) A1

New Resistance: R2=ρL2A2=ρ⋅2L1/ (1/2)A=2ρL1 / (1/2)A1=(2ρL1×2)/A1=4ρL1/ A1

​So, the new resistance R2​ becomes:

R2=4R1

Therefore, the correct option is:

B. Four times

(Option B) Four times: The calculation shows that when the length is doubled and the area is halved, the resistance increases by a factor of four. This is the correct answer.

(Option A) Double: If the resistance doubled, it would mean the new resistance R2​ is 2R1​. This would happen if only the length doubled while keeping the area the same, which is not the case here.

(Option C) One half: If the resistance was halved, R2​ would be R1/ 2​​. This could happen if the length remained the same but the area doubled, which again is not the scenario here.

(Option D) One fourth: If the resistance became one fourth, R2​ would be R1/ 4​​. This would require the length to be halved and the area to be doubled, which does not match our given conditions.

Subject: Physics

Topic: Current Electricity

Subtopic: Resistivity and its dependence upon temperature

Difficulty Level: Hard