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Chapter 1: Problem 2

Choose the correct statements from the following: (1) The general form of Newton's second law of motion is \(\vec{F}_{\mathrm{ext}}=m \vec{a}\) (2) A body can have energy and get no momentum. (3) A body having momentum must necessarily have kinetic energy. (4) The relative velocity of two bodies in a head-on elastic collision remains unchanged in magnitude and direction.

Short Answer

Expert verified
The correct statements are 1, 2, and 3.

Step by step solution

01

Understanding Statement 1

Statement 1 presents Newton's second law of motion. The general form of Newton's second law is indeed given by \(\vec{F}_{\mathrm{ext}} = m \vec{a}\), where \(\vec{F}_{\mathrm{ext}}\) is the external force, \(m\) is the mass of the object, and \(\vec{a}\) is the acceleration. This statement is correct.
02

Analyzing Statement 2

Statement 2 claims a body can have energy and get no momentum. A body can have potential energy, such as when it is held at a height, without having momentum. This is because potential energy depends on position, not motion, unlike momentum, which depends on velocity. Therefore, this statement is correct.
03

Evaluating Statement 3

Statement 3 suggests a body having momentum must necessarily have kinetic energy. Since momentum is given by \(\vec{p} = m\vec{v}\) and kinetic energy by \(KE = \frac{1}{2}mv^2\), any non-zero velocity \(\vec{v}\) implies both non-zero momentum and kinetic energy. Thus, any object with momentum must have kinetic energy. This statement is correct.
04

Examining Statement 4

Statement 4 refers to the relative velocity in a head-on elastic collision. In such collisions, the magnitude of the relative velocity of the two bodies remains unchanged, but the direction generally reverses, not remains unchanged. Therefore, this statement is incorrect.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

momentum
Momentum is a key concept in physics that refers to the quantity of motion an object has. It is defined as the product of an object's mass and its velocity, expressed by the equation \[\vec{p} = m\vec{v}\]where \(\vec{p}\) denotes momentum, \(m\) is the mass, and \(\vec{v}\) is the velocity of the object.

Some important things to remember about momentum include:
  • It is a vector quantity, meaning it has both magnitude and direction.
  • It depends directly on both the mass and velocity of an object. Increasing either will increase momentum.
  • In a closed system, total momentum is conserved, which means that momentum lost by one object is gained by another.
Understanding momentum is essential for solving problems related to collisions and systems of particles.
elastic collision
An elastic collision is a type of collision where both momentum and kinetic energy are conserved. This means that the total kinetic energy of the system before the collision is equal to the total kinetic energy after the collision.

Characteristics of elastic collisions:
  • No kinetic energy is lost. Energy is not transformed into other forms like sound or heat.
  • The relative velocity of the colliding bodies is the same before and after the impact, but the directions may change.
  • Commonly observed in atomic and subatomic particle interactions but less so in macroscopic objects.
Elastic collisions serve as idealized models that can greatly simplify analyzing collision problems, especially in physics exercises.
potential energy
Potential energy is the stored energy in an object due to its position or state. It is a scalar quantity, which means it has magnitude but no direction.

Forms of potential energy include:
  • Gravitational potential energy: Energy due to an object's position relative to the Earth, calculated by\[ PE = mgh \]where \(m\) is mass, \(g\) is the acceleration due to gravity, and \(h\) is height above the ground.
  • Elastic potential energy: Energy stored in objects like springs, calculated by\[ PE = \frac{1}{2} k x^2 \]where \(k\) is the spring constant and \(x\) is the displacement from the equilibrium position.
Potential energy is a versatile concept applied in various areas like mechanics, quantum mechanics, and thermodynamics.
kinetic energy
Kinetic energy refers to the energy an object possesses due to its motion. It is given by the formula: \[ KE = \frac{1}{2}mv^2 \]where \(KE\) stands for kinetic energy, \(m\) is the mass, and \(v\) is the velocity of the object.

Key features of kinetic energy:
  • It is a scalar quantity, which means it does not have a direction, only magnitude.
  • Kinetic energy increases with the square of the velocity, meaning that doubling the speed of an object results in quadrupling its kinetic energy.
  • Unlike potential energy, kinetic energy directly relates to the motion of the object.
Kinetic energy is fundamental in analyzing moving objects and is paramount in energy conservation problems.
relative velocity
Relative velocity is a concept that describes the velocity of one object as observed from another moving object. It is useful in understanding how objects are moving about each other in a frame of reference.

Key points about relative velocity:
  • It is calculated by subtracting the velocity vector of one object from the velocity vector of another,\(\vec{v}_{\text{relative}} = \vec{v}_A - \vec{v}_B\) where \(\vec{v}_A\) is the velocity of the first object and \(\vec{v}_B\) is the velocity of the second object.
  • In a head-on collision, the relative velocity between two objects is initially negative if they are moving towards each other, and positive afterward if they reverse direction.
  • This concept is essential in collision problems, allowing for the analysis of how velocities change during interactions.
Understanding relative velocity helps in solving complex motion problems, especially those involving more than one moving object.

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Most popular questions from this chapter

A massless spring of force constant \(1000 \mathrm{Nm}^{-1}\) is compressed a distance of \(20 \mathrm{~cm}\) between discs of \(8 \mathrm{~kg}\) and \(2 \mathrm{~kg}\), spring is not attached to discs. The system is given an initial velocity \(3 \mathrm{~ms}^{-1}\) perpendicular to length of spring as shown in the figure. What is ground frame velocity of \(2 \mathrm{~kg}\) block (in \(\mathrm{ms}^{-1}\) ) when spring regains its natural length.

Two identical carts constrained to move on a straight line on which sit two twins of same mass, are moving with equa velocity. At some time snow begins to drop uniformly. Ram sitting on one of the carts, picks the snow from cart an throws off the falling snow sideways and in the second can Shyam is asleep. (1) Cart carrying Ram will have more speed finally than the carrying Shyam. (2) Cart carrying Ram will have less speed finally than thel carrying Shyam. (3) Cart carrying Ram will have same speed finally than the carrying Shyam. (4) depends on the amount of snow thrown.

A trolley is moving horizontally with a velocity of \(v \mathrm{~m} / \mathrm{s}\) w.r.t. earth. A man starts running in the direction of motion of trolley from one end of the trolley with a velocity \(1.5 v\) \(\mathrm{m} / \mathrm{s}\) w.r.t. the trolley. After reaching the opposite end, the man turns back and continues running with a velocity of \(1.5\) \(v \mathrm{~m} / \mathrm{s}\) w.r.t. trolley in the backward direction. If the length of the trolley is \(L\), then the displacement of the man with respect to earth, measured as a function of time, will attain a maximum value of (1) \(\frac{4}{3} L\) (2) \(\frac{2}{3} L\) (3) \(\frac{5 L}{3}\) (4) \(1.5 L\)

A frog sits on the end of a long board of length \(L=5 \mathrm{~m}\). The board rests on a frictionless horizontal table. The frog wants to jump to the opposite end of the board. What is the minimum take-off speed (in \(\mathrm{m} / \mathrm{s}\) ), i.e., relative to ground that allows the frog to do the trick? The board and the frog have equal masses.

A man standing on the edge of the terrace of a high rise building throws a stone, vertically up with a speed of \(20 \mathrm{~m} / \mathrm{s}\). Two seconds later, an identical stone is thrown vertically downwards with the same speed of \(20 \mathrm{~m} / \mathrm{s}\). Then (1) the relative velocity between the two stones remains constant till one hits the ground (2) both will have the same kinetic energy, when they hit the ground (3) the time interval between their hitting the ground is \(2 \mathrm{~s}\) (4) if the collision on the ground is perfectly elastic, both will rise to the same height above the ground
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