Vector Dot Product Quiz

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Questions and Answers

What does the scalar product of two vectors represent geometrically?

  • The difference between the two vector magnitudes
  • The sum of the magnitudes of the vectors
  • The magnitude of the vectors multiplied by the angle between them
  • The projection of one vector onto another (correct)

What is the value of the scalar product $ ilde{i} ullet ilde{j}$?

  • 0 (correct)
  • Undefined
  • 1
  • -1

If the angle $ heta$ between two vectors $ ilde{A}$ and $ ilde{B}$ is 90 degrees, what is the value of $ ilde{A} ullet ilde{B}$?

  • 1
  • 0 (correct)
  • Undefined
  • The product of their magnitudes

How would increasing the angle $ heta$ between two vectors affect their dot product?

<p>Decrease the dot product (C)</p> Signup and view all the answers

What is the result of $ ilde{j} ullet ilde{k}$?

<p>0 (A)</p> Signup and view all the answers

Which of the following statements about the scalar product is true?

<p>It measures how much one vector extends in the direction of another (C)</p> Signup and view all the answers

Which form of energy results from work done by friction?

<p>Heat Energy (B)</p> Signup and view all the answers

What does the coefficient of restitution (e) signify in collision theory?

<p>The elasticity of a collision (B)</p> Signup and view all the answers

In an inelastic collision, which of the following remains constant?

<p>Linear momentum (A)</p> Signup and view all the answers

According to the Law of Conservation of Energy, energy can be:

<p>Transformed into another form (A)</p> Signup and view all the answers

What is the correct formula for calculating work done by a constant force?

<p>$W = F * s$ (A)</p> Signup and view all the answers

What equation represents work in terms of power?

<p>$W = P imes t$ (D)</p> Signup and view all the answers

How is kinetic energy calculated?

<p>$K = rac{1}{2}mv^2$ (A)</p> Signup and view all the answers

Which energy form arises from the flow of electrical current?

<p>Electrical Energy (D)</p> Signup and view all the answers

What does the Work-Energy Theorem state?

<p>The change in kinetic energy is equal to the net work done on an object. (D)</p> Signup and view all the answers

What is the value of the coefficient of restitution for a perfectly inelastic collision?

<p>0 (C)</p> Signup and view all the answers

When is work considered positive?

<p>When the force and displacement are in the same direction. (C)</p> Signup and view all the answers

Which formula corresponds to Hooke's Law?

<p>$F = -kx$ (D)</p> Signup and view all the answers

Which form of energy arises due to the nuclear reactions of fission or fusion?

<p>Nuclear Energy (D)</p> Signup and view all the answers

In the context of variable forces, which equation is used to calculate work done?

<p>$W = ext{integral}_{x_1}^{x_2} F(x) , dx$ (A)</p> Signup and view all the answers

What happens to kinetic energy when negative work is done on an object?

<p>Kinetic energy decreases. (B)</p> Signup and view all the answers

What does the equation $E = mc^2$ illustrate?

<p>The relationship between mass and energy (D)</p> Signup and view all the answers

Which of the following statements is true about variable forces?

<p>Work done requires integration to calculate for variable forces. (C)</p> Signup and view all the answers

Which of the following equations relates momentum to kinetic energy?

<p>$K = rac{p^2}{2m}$ (D)</p> Signup and view all the answers

What is the condition for work to be done on a body?

<p>There must be both a force and corresponding displacement. (B)</p> Signup and view all the answers

Which best describes negative work?

<p>Force and displacement are in opposite directions. (D)</p> Signup and view all the answers

What is the characteristic of conservative forces?

<p>The work done depends only on initial and final positions. (D)</p> Signup and view all the answers

How is gravitational potential energy expressed?

<p>$V(h) = mgh$ (A)</p> Signup and view all the answers

What does the work-energy theorem state?

<p>The change in kinetic energy is equal to the net work done. (C)</p> Signup and view all the answers

What is the formula for tension when an object moves in a vertical circle?

<p>$T = rac{mv^{2}}{r} + mg ext{cos} heta$ (D)</p> Signup and view all the answers

What determines the minimum velocity for an object at the highest point of a vertical circle?

<p>The speed must be sufficient to maintain tension in the string. (B)</p> Signup and view all the answers

What is the relationship between potential energy and height for an object in gravitational fields?

<p>Potential energy increases linearly with height. (B)</p> Signup and view all the answers

If an object moves in a vertical circle, what happens to its kinetic and potential energy at the lowest point?

<p>Kinetic energy is maximum, potential energy is minimum. (A)</p> Signup and view all the answers

How is the minimum velocity required for an object to complete a vertical loop determined?

<p>It depends solely on the radius of the loop. (C)</p> Signup and view all the answers

What mathematical representation defines potential energy concerning force?

<p>$F(x) = - rac{dV}{dx}$ (C)</p> Signup and view all the answers

What does the Work-Energy Theorem primarily explain?

<p>The relationship between work done on an object and its change in kinetic energy (D)</p> Signup and view all the answers

In elastic collisions, which of the following is conserved?

<p>Kinetic energy and momentum (D)</p> Signup and view all the answers

When analyzing work done by gravity on an object, which factor is crucial for the calculation?

<p>The angle of displacement relative to the force of gravity (B)</p> Signup and view all the answers

Which equation represents the kinetic energy of an object?

<p>KE = rac{1}{2}mv^2 (B)</p> Signup and view all the answers

What is the primary focus of vector analysis involving dot products?

<p>Finding the angle between two vectors (D)</p> Signup and view all the answers

In inelastic collisions, what occurs to kinetic energy?

<p>Some of it is lost to other forms of energy (D)</p> Signup and view all the answers

What is the effect of friction on the work done on an object?

<p>Decreases the work done on the object (B)</p> Signup and view all the answers

When calculating the work done by a force, what is the formula involving force and displacement?

<p>Work = Force × Distance × cos(θ) (B)</p> Signup and view all the answers

Which scenario best illustrates the concept of momentum conservation?

<p>Two ice skaters pushing off of each other (D)</p> Signup and view all the answers

What role does gravitational force play in the work-energy relationship?

<p>It can do work depending on the motion of the object (B)</p> Signup and view all the answers

Flashcards

Conservative Force

A force where the work done only depends on the starting and ending positions, and the total work done in a closed path is zero.

Potential Energy (V(x))

A scalar quantity that describes the energy stored in a system due to its position. The force can be derived from the negative derivative of the potential energy function

Work-Energy Theorem

The change in kinetic energy of an object is equal to the net work done on it.

Gravitational Potential Energy

The energy stored in an object due to its height above a reference point.

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Vertical Circle Motion

The motion of an object moving in a circular path that is vertical.

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Minimum Velocity (Vertical Circle)

The smallest speed required for an object to complete a vertical circular path without falling.

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Tension in Vertical Circle

The force exerted by a string or rope keeping an object in a curved path. At any point, T=mv²/r+mgcosθ

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Work (constant force)

Force multiplied by distance in the direction of the force.

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Work (variable force)

Calculated by integrating the force over the distance.

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Kinetic Energy

Energy of motion, measured by 1/2 * mass * velocity^2.

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Work-Energy Theorem

Change in kinetic energy equals net work done.

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Variable Force

Force that changes strength or direction.

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Positive Work

Force and displacement are in the same direction.

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Negative Work

Force and displacement are in opposite directions.

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Forms of Energy

Energy exists in various forms, including heat, chemical, electrical, and nuclear energy.

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Law of Conservation of Energy

Energy cannot be created or destroyed, only transformed.

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Elastic Collision

Collision where momentum and kinetic energy are conserved.

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Inelastic Collision

Collision where kinetic energy is not conserved, but momentum is.

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Coefficient of Restitution (e)

Measures the elasticity of a collision; e=1 is perfectly elastic, e=0 is perfectly inelastic.

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Collision

An event where two bodies interact or change path due to interaction.

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Hooke's Law

The force exerted by a spring is proportional to its displacement.

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Mass-energy equivalence

E=mc²; Mass and energy are equivalent, and can convert in different amounts.

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Instantaneous Power

Power at a specific moment in time calculated as rate of work done.

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Work-Energy Theorem

The change in kinetic energy of an object equals the net work done on it.

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Kinetic Energy

Energy of motion, calculated by 1/2 * mass * velocity^2

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Elastic Collision

Collision where kinetic energy is conserved.

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Inelastic Collision

Collision where kinetic energy is not conserved.

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Work by Force

Force applied over a distance; measured in Joules.

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Vector Analysis

Calculations involving quantities with magnitude and direction.

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Scalar Product

The dot product of two vectors, resulting in a scalar value. It's calculated by multiplying the magnitudes of the vectors and the cosine of the angle between them.

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Dot Product Formula

A ⋅ B = |A| |B| cos θ, where A and B are vectors, |A| and |B| are their magnitudes, and θ is the angle between them.

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Projection of a Vector

The component of a vector along another vector. In scalar product, |B| cos θ is the projection of vector B onto vector A.

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Unit Vector

A vector with a magnitude of 1. Used for directions along an axis.

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i â‹… i

1

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i â‹… j

0

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