College Physics I: BIIG problem-solving method

Created Feb. 7, 2024 by C N Hiremath

Two-dimensional Kinematics

  • Examples of motions along curved paths: The orbit of a satellite; The arc of a basketball; A swimmer diving into a pool
  • Both two- and three-dimensional kinematics are simple extensions of the one-dimensional kinematics
  • This allows us to apply physics to many more situations, and it will also yield unexpected insights about nature.

 

Pythagorean Theorem

  • The Pythagorean theorem relates the length of the legs of a right triangle, labeled a and b, with the hypotenuse, labeled c.  The relationship is given by:

a2  +  b2  =  c2

This can be rewritten, solving for c:

c   =  √ ( a2 + b2 )

 

Independence of Motion

  • The horizontal and vertical components of two-dimensional motion are independent of each other.

Any motion in the horizontal direction does not affect motion in the vertical direction, and vice versa.

  • Assuming no air resistance, the vertical motion of a falling object is influenced by gravity only, and not by any horizontal forces.
  • The length of the arrow is proportional to the vector’s magnitude. The arrow points in the same direction as the vector.

 

Resolving Vectors

  • The key to analyzing such motion, called projectile motion, is to resolve (break) it into motions along perpendicular directions.
  • Resolving two-dimensional motion into perpendicular components is possible because the components are independent.

 

Vectors in Two Dimensions

  • A vector is a quantity that has magnitude and direction.
  • Displacement, velocity, acceleration, and force, for example, are all vectors.

 

Vector Addition

  • The head-to-tail method is a graphical way to add vectors. The tail of the vector is the starting point of the vector, and the head (or tip) of a vector is the final, pointed end of the arrow.
  • The negative of a vector B is defined to be –B; that is, graphically the negative of any vector has the same magnitude but the opposite direction. Essentially, we just flip the vector so it points in the opposite direction.

 

Vector Subtraction

  • Vector subtraction is a straightforward extension of vector addition.
  • The subtraction of vector B from vector A is then simply defined to be the addition of –B to A.
  • Note that vector subtraction is the addition of a negative vector.

            A – B  =  A +  (–B)

Multiplication of Vectors

  • When vector A is multiplied by a scalar c, the magnitude of the vector becomes the absolute value of

c A

if c is positive, the direction of the vector does not change,

if c is negative, the direction is reversed.

 

Resolving a Vector into Components

  • Given a vector like A, the two perpendicular vectors, Ax and Ay are its x- and y-components, respectively. These vectors form a right triangle.

             Ax + Ay = A

  • Note that this relationship between vector components and the resultant vector holds only for vector quantities.
  • The magnitudes of the vector components Ax and Ay can be related to the resultant vector A and the angle θ with trigonometric identities.

             Ax = A cos θ      and       Ay = A sin θ

  • Calculating the resultant vector: The magnitude A and the direction θ are

            A  =  √ ( Ax2 + Ay2 )     and           θ  =  tan−1 ( Ay / Ax )

  • The components of the resultant along each axis are obtained by adding the components of the individual vectors along that axis.

            Rx  =  Ax + Bx             and           Ry  =  Ay + By

 

Projectile Motion

  • Projective motion is the motion of an object thrown or projected into the air, subject to only the acceleration of gravity. The object is called a projectile, and its path is called its trajectory.
  • The motion of falling objects, is a simple one-dimensional type of projectile motion in which there is no horizontal movement.
  • The most important fact to remember here is that motions along perpendicular axes are independent and thus can be analyzed separately.
  • For horizontal motion ( ax = 0 )

x  =  x0  +  vx t

vx =  v0x  =  vx  = velocity is a constant

  • For vertical motion ( ay = −g = −9.80m/s2 )

vy =  v0y − g t

y  =  y0 + v0y t − ½ g t2

vy2 =  v0y2 − 2 g ( y − y0 )

  • Problem (P3.4): During a fireworks display, a shell is shot into the air with an initial speed of 70.0 m/s at an angle of 75.0º above the horizontal. The fuse is timed to ignite the shell just as it reaches its highest point above the ground. Calculate the height at which the shell explodes.                     ( 233 m )
  • When air resistance is negligible, the maximum height  y = h,

h =   v0y2  / 2 g

The maximum height of a projectile depends only on the vertical component of the initial velocity.

  • The range R of a projectile on level ground for which air resistance is negligible is given by

            R = v02 sin 2 θ0 / g

Where, v0 is the initial speed and  θ0 is the initial angle relative to the horizontal.

  • For a fixed initial speed,the maximum range is obtained with θ0 = 45º

 

 

Relative Velocity

  • Examples of relative velocity: A boat moving across a rapidly flowing river; A small airplane flying in a strong crosswind.
  • The rules of vector addition and subtraction apply!
  • Problem (P3.7): The plane is known to be moving at 45.0 m/s due north relative to the air
    mass, while its velocity relative to the ground (its total velocity) is 38.0 m/s in a direction 20.0º west of north. Calculate is the speed and direction of the wind.                                             (16.0 m/s  &  36.6º SW )

 

Relative Velocities & Classical Relativity

  • When adding velocities, we have been careful to specify that the velocity is relative to some reference frame. These velocities are called relative velocities.
  • For example, the velocity of an airplane relative to an air mass is different from its velocity relative to the ground.
  • Relative velocities are one aspect of relativity, which is defined to be the study of how different observers moving relative to each other measure the same phenomenon.
  • Classical relativity is limited to situations where speeds are less than about 1% of the speed of light that is, less than 3,000 km/s. Most things we encounter in daily life move slower than this speed.
  • Einstein revolutionized our view of nature with his modern theory of relativity.
  • Problem (P3.8): An airline passenger drops a coin while the plane is moving at 260 m/s. What is the velocity of the coin when it strikes the floor 1.50 m below its point of release: Measured relative to the plane? Measured relative to the Earth?                                                                                        ( 260 m/s ) 

 

BIIG: Problems & Solutions


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