Chapter 3 motion in 2 or 3 dimensions
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Chapter 3: Motion in 2 or 3 Dimensions. Position & Velocity Vectors. Position Vector. To describe the motion of a particle in space, we first need to describe the position of the particle.

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Position Vector

  • To describe the motion of a particle in space, we first need to describe the position of the particle.

  • Position vector of a particle is a vector that goes from the origin of coordinate system to the point.

  • Position vector components are the Cartesian coordinates of the particle.

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Position & Velocity Vectors

  • As the particle moves through space, the path is a curve.

  • The change in position (the displacement) of a particle during time interval t is:

  • Average velocity vector during this time interval is the displacement divided by the time interval:

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Position & Velocity Vectors

  • Instantaneous velocity vector is the limit of the average velocity as the time interval approaches zero, and equals the instantaneous rate of change of position with time:

  • Magnitude of the vector v at any instant is the speedv of the particle at that instant.

  • Direction of v at any instant is the same as the direction in which particle moves at that instant.

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Position & Velocity Vectors

  • As t0, P1 and P2 move closer and in this limit vector r becomes tangent to the curve.

  • Direction of r in the limit is the same as direction of instantaneous velocity v.

  • At every point along the path, instantaneous velocity vector v is the tangent to the path at that point.

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Position & Velocity Vectors

  • Components of instantaneous velocity vector v :

  • Magnitude of vector v by Pythagorean theorem:

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Acceleration Vector

  • Acceleration of a particle moving in space describes how the velocity of particle changes.

  • Average acceleration is a vector change in velocity divided by the time interval:

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Acceleration Vector

  • Instantaneous acceleration is the limit of the average acceleration as the time interval approaches zero, and equals the instantaneous rate of change of velocity with time:

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Acceleration Vector

  • In terms of unit vectors:

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Acceleration Vector

  • Components of acceleration:

  • Acceleration vector can be resolved into a component parallel to the path (and velocity), and a component perpendicular to the path.

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Acceleration Vector

  • Components of acceleration:

  • When accelerationvector is parallel to the path (and velocity), the magnitude of v increases, but its direction doesn’t change

  • When accelerationvector is perpendicular to the path (and velocity), the direction of v changes, but magnitude is constant

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Acceleration Vector

  • Components of acceleration for a particle moving along a curved path:

  • Constant speed

  • Increasing speed

  • Decreasing speed

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Projectile Motion

  • A projectile is any object that is given an initial velocity and then follows a path (trajectory) determined solely by gravity and air resistance.

  • The motion of a projectile will take place in a plane (so, it is 2-D motion).

  • For projectile motion we can analyze the x- and y-components of the motion separately.

  • The horizontal motion (along the x-axis) will have zero acceleration and thus have constant velocity.

  • The vertical motion (along the y-axis) will have constant downward acceleration of magnitude g = 9.80 m/s2.

  • The initial velocity components, vox and voy, can be expressed in terms of the magnitude vo and direction ao of the initial velocity.

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Projectile Motion

  • We analyze projectile motion as a combination of horizontal motion with constant velocity and vertical motion with constant acceleration

  • Initial velocity is represented by its magnitude and direction

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Projectile Motion

  • Trajectory of a body projected with initial velocity v0

  • h is maximum height of trajectory

  • R is horizontal range

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Projectile Motion

  • Trajectory of a cow

  • A cow is launched from the top of a hill with an initial velocity vector that makes an angle of 45 degrees with the horizontal. The projectile lands at a point that is 10 m vertically below the launch point and 300 m horizontally away from the launch point.

  • Determine the time the cow was in the air.

  • Determine the initial speed of the cow.

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Projectile Motion

  • Initial speed of the cow

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Projectile Motion

  • Flight time of the cow

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Motion in a Circle

  • When a particle moves along a curve, direction of its velocity vector changes.

  • Particle musthave component of acceleration to the curved path even if the speed is constant.

  • Motion in a circle is a special case of motion along a curved path.

  • Uniform circular motion - when a particle moves with constant speed

  • Non-uniform circular motion - if the speed of a particle varies.

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Uniform Circular Motion

  • No component of acceleration parallel (tangent) to the path. Otherwise, speed would change.

  • Non-zero component of acceleration is perpendicular to the path.

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Uniform Circular Motion

  • A particle that is undergoing motion in such a manner that its direction is changing is experiencing a radial acceleration that has magnitude equal to the square of its velocity divided by the instantaneous radius of curvature of its motion. The direction of this radial, or centripetal, acceleration is toward the center of circular path of particle's motion.

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Uniform Circular Motion

  • In uniform circular motion, the magnitude a of instantaneous acceleration is equal to the square of the speed v divided by the radius R of the circle.

  • Its direction is  to v and inward along the radius.

  • Centripetal “seeking the center” (Greek)

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Uniform Circular Motion

  • Periodof the motion T the time of one revolution, or one complete trip around the circle.

  • In time T, particle travels the distance 2Rof the circle, so its speed can be expressed as

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Motion in a Circle: Example

Centripetal acceleration on a curved road

  • A car has a “lateral acceleration” of 0.87g, which is (0.87)/(9.8m/s2)=8.5m/s2. This represents the maximum centripetal acceleration that can be attained without skidding out of the circular path. If the car is traveling at a constant speed 40m/s (~89mi/h, or 144km/h), what is the max radius of curve it can negotiate?


  • Car travels along a curve, speed is constant  apply equation of circular motion to find the target variable R.


  • We know arad and v, so we can findR:

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Non-Uniform Circular Motion

  • An object that is undergoing non-uniform circular motion, or motion where the magnitude and the direction of the velocity is changing, will experience an acceleration that can be described by two components:

  • A radial or centripetal acceleration equal to the square of speed divided by radius of curvature of motion directed toward the center of curvature of the motion, and

  • Tangential component of acceleration that is equal to the rate of change of the particle's speed and is directed either parallel (in the case of speeding up) or anti-parallel (in the case of slowing down) to the particle's velocity.

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Relative Velocity

  • The velocity seen by particular observer is called relative to that observer, or relative velocity.

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Relative Velocity in 1-D

  • Woman walks with a velocity of 1.0m/s along the aisle of a train that is moving with a velocity of 3.0m/s. What is the woman’s velocity?

  • For passenger sitting in a train: 1.0m/s

  • For bicyclist standing: 1.0m/s + 3.0m/s = 4.0m/s

  • Frame of reference is a coordinate system + time scale

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Relative Velocity in 1-D

  • Cyclist: frame of reference A

  • Moving train: frame of reference B

  • In 1-D motion, position of P relative to frame of reference A is given by distance XP/A

  • Position of P relative to frame of reference B is given by distance XP/B

  • Distance from origin A to origin B is given by XB/A

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Relative Velocity in 1-D

  • Velocity VP/A of P relative to frame A is the derivative of XP/A with respect to time