College Physics I: BIIG problem-solving method

Created Feb. 7, 2024 by C N Hiremath

Kinematics

  • Kinematics is a modern name for the mathematical description of motion.
  • Kinematics is defined as the study of motion without considering its causes.
  • It comes from a Greek term meaning motion.

 

Motion

  • The change of an object’s position or orientation with time.
  • Four basic types of motion:

Straight-line motion: Motion along a straight line

Circular motion: Motion along a circular path

Projectile motion: Motion of an object through the air

Rotational motion: Spinning of an object about an axis

 

Position

  • Position is describing where an object is at any particular time.
  • It is specified relative to a convenient reference frame.
  • For example, a plane at the start of a runway during take-off; a person walking towards the front of the moving bus.

 

Distance

  • Distance is defined to be the magnitude or size of displacement between two positions.
  • It has no direction and, thus no sign.
  • Distance traveled is the total length of the path traveled between two positions.
  • The distance between two positions is NOT the same as the distance traveled between them.

 

Displacement

  • Displacement is change in position of an object.
  • It is described in terms of direction.
  • If x0 is the initial position, and xf is the final position, then change in position is given by displacement

Δx  =  xf  - x0

  • The SI unit for displacement is the meter (m).

 

Vectors and Scalars

  • A vector is any quantity with both magnitude and direction.

The direction of a vector in one-dimensional motion is given simply by a plus (+) or minus (-).

Vectors are represented graphically by arrows. An arrow used to represent a vector has a length proportional to the vector’s magnitude.

  • A scalar is any quantity that has a magnitude, but no direction. Scalars are never represented by arrows. A scalar can be negative.

 

Coordinate Systems

  • In order to describe the direction of a vector quantity, we must designate a coordinate system within the reference frame.

 

Time

  • Every measurement of time involves measuring a change in some physical quantity.
  • Time is change, or the interval over which change occurs.
  • The SI unit for time is the second (s).
  • Elapsed time or time interval

Δt  =  tf  - t0

  • Is the difference between the ending time and beginning time.
  • For simplicity,  

Δt  =  tf   ≡  t

 

Velocity

  • Average velocity is displacement (change in position) divided by the time of travel.

v-  =  ∆x / ∆t  =   ( xf  - x0 ) / ( tf  - t0 )

  • The SI unit for velocity is meters per second or m/s.
  • Other common units are km/h, mi/h (mph), and cm/s.
  • Instantaneous velocity, v is the average velocity at a specific instant in time (or over an infinitesimally small time interval).
  • It involves taking a limit, a calculus operation beyond the scope of this course.

 

Acceleration

  • Average acceleration is the rate at which velocity changes.

a-  =  ∆v / ∆t  =   ( vf  - v0 ) / ( tf  - t0 )

  • The SI unit for velocity is meters per second squared or meters per second per second or m/s2.
  • Acceleration vector is a vector in the same direction as the change in velocity, ∆v.
  • Acceleration is therefore a change in either speed or direction, or both.
  • Deceleration is when an object slows down, its acceleration is opposite to the direction of its motion.
  • Problem (P2.1): A racehorse coming out of the gate accelerates from rest to a velocity of 15.0 m/s due west in 1.80 s. What is its average acceleration?                                                      ( - 8.33 m/s2 )
  • Instantaneous acceleration  a, or the acceleration at a specific instant in time,

It is obtained by considering an infinitesimally small interval of time.

  • Problem (P2.5): At the end of its trip, the train slows to a stop from a speed of 30.0 km/h in 8.00 s. What is its average acceleration while stopping?                                                                     ( - 1.04 m/s2 )

 

Kinematic Equations

  • Summary of Kinematic Equations (constant acceleration)

Final position              x  =  x0 + v- t

            Average velocity         v-  =  (v0  + v) / 2

            Final velocity              v  =  v0 + a t

            Final position              x  =  x0 + v0 t + ½ a t2

            Final velocity square   v2 =  v02  + 2 a ( x - x0 )

  • Problem (P2.9): An airplane lands with an initial velocity of 70.0 m/s and then decelerates at 1.50 m/s2 for 40.0 s.  What is its final velocity?                                                                   ( 10.0 m/s )
  • Problem (P2.13): Suppose a car merges into freeway traffic on a 200-m-long ramp. If its initial velocity is 10.0 m/s and it accelerates at 2.00 m/s2. How long does it take to travel the ramp?  ( 10 s )

 

Falling Objects

  • If air resistance and friction are negligible, then all objects fall toward the center of Earth with the same constant acceleration, independent of their mass.
  • An object falling without air resistance or friction is defined to be in free-fall.

 

Acceleration due to Gravity

  • The force of gravity causes objects to fall toward the center of Earth.
  • The acceleration due to gravity is the acceleration of free-falling objects.
  • Its magnitude is given by the symbol, g.  Depending on latitude, altitude, underlying geological formations, and local topography, g varies from

 9.78 m/s2     to      9.83 m/s2

  • It is constant at any given location on Earth and has the average value

g  =  9.80 m/s2

  • The direction of the acceleration due to gravity is downward (toward the center of Earth).
  • The acceleration a in the kinematic equations has the value +g or –g depending on how the coordinate system is defined.

 

One-Dimensional Motion involving Gravity

  • Kinematic equations for objects in free-fall where acceleration,  a  =   -g  and the vertical displacement is represented by the symbol y

v  =  v0 - g t

y  =  y0 + v0 t - ½ g t2

v2 =  v02  - 2 g ( y - y0 )

  • Problem (P2.14): A person standing on the edge of a high cliff throws a rock straight up with an initial velocity of 13.0 m/s.  The rock misses the edge of the cliff as it falls back to earth.  Calculate the position and velocity of the rock 2.00 s after it is thrown, neglecting air resistance.          ( - 6.60  m/s )

 

Graphical Analysis

  • Graphs not only contain numerical information; they also reveal relationships between physical quantities.
  • The graph of displacement x vs. time. This is kinematic equation for final position.

x  =  v- t  +  x0             

The slope is velocity v

slope  =  Δx / Δt  =  v  

  • The graph of velocity v vs. time t. This is kinematic equation for final velocity.

v  =  a t  +  v0             

The slope is acceleration a

slope  =  Δv / Δt  =  a


BIIG: Problems & Solutions


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