Make your own free website on Tripod.com
Home Page of Peggy E. Schweiger

MOTION IN TWO DIMENSIONS

projectile
any object thrown or otherwise projected into the air
trajectory
the parabolic path of a projectile

In projectile motion, the horizontal and the vertical components of the motion are treated separately. A projectile moves both horizontally and vertically. Its horizontal motion is constant. Its vertical motion is affected by the acceleration due to gravity. The only variable shared by both types of motion is time. Every point on the trajectory is the vector sum of the horizontal and the vertical components of the velocity.

Speed

The speed of an object at any point on the trajectory can be found by calculating the horizontal and vertical velocity components at that point. The speed is the vector sum of the components at that point.

An object projected horizontally (projected perfectly parallel to the surface) will reach the ground in the same time as an object dropped vertically. Since speed at any point in a trajectory is the vector sum of the horizontal and vertical velocity components at that point, the projected object will have a greater speed when it strikes.

The maximum range for a given initial velocity is obtained when the angle of projection is 45°.

Equations that are used to describe the horizontal and vertical motion:

Directions

It is very important to consistently define directions in projectile motion. If the acceleration due to gravity is defined to be negative, then all velocities in the "down" direction are also negative. If displacement is measured from the ground up, it is positive. If it is measured from the "top" down, it is negative.

Advanced calculations

The general equation of motion, d = vit + a t2, can be easily used to calculate vertical displacement and/or time at any point in a trajectory. First, find the initial vertical velocity component.

We will be working two types of problems.

A formula can be derived for the horizontal range:

Range = (v2 sin 2q) / g
where v is the projectile velocity, q is the projectile angle, and g is the acceleration due to gravity.

Advanced calculations

The general equation of motion, d = vit + a t2, can be easily used to calculate vertical displacement and/or time at any point in a trajectory. First, find the initial vertical velocity component.

A virtual lab in which the user can control the angle of the cannon. Cursor interrogation allows maximum vertical height and total horizontal range to be determined. A target is provided for amusement.

These two types of problems can also be solved graphically using parametric equations on the graphing calculator. If you are interested in learning how to do this, it is explained in the link below.

Graphing calculator solution to problems

One of the classic questions in physics is this: What should a monkey in a tree do when a gun pointed at him fires -- jump down or stay where it is? A graphing calculator solution to this problem is found at Monkey and Hunter Problem

SIMPLE HARMONIC MOTION

simple harmonic motion
   periodic motion where the unbalanced force varies directly with the displacement from the equilibrium point; this motion is described by the period, the frequency, and the amplitude of the motion

period (T)
    the time in seconds needed to complete one cycle of motion

amplitude
    the distance from the equilibrium point to the point of greatest displacement

frequency
    the number of vibrations in a time interval; its SI unit is hertz (Hz)

I like to use the Greek letter, n, as the symbol for frequency. This can be confusing to students since its appearance is similar to the letter v. It can also be represented using the symbol f

1 Hz = 1 sec-1
T = 1/n  or T = 1/f
n = 1/T  or f = 1/T

characteristics of a simple pendulum:

  1. period is independent of mass
  2. period is directly proportional to the square root of its length
  3. period is indirectly proportional to the square root of the acceleration due to gravity
  4. period is independent of amplitude if the arc is less than 10°

formula for period of pendulum

Where l is the length of the pendulum and g is the acceleration due to gravity at that point.

For a pendulum, speed is zero and acceleration is a maximum at the point of maximum displacement (point A). For a pendulum, speed is a maximum and acceleration is zero at the equilibrium point (point B).

cycle of a pendulum

UNIFORM CIRCULAR MOTION

acceleration involves a change in speed and/or direction; it is caused by an unbalanced force

in circular motion, the object moves at constant speed but is accelerating because its direction is constantly changing

Uniform Circular Motion

An object moving in a circle of radius r with constant speed v has an acceleration whose direction is toward the center of the circle adn whose magnitude is aR = v 2/r. Acceleration depends upon speed and radius. The greater the speed, the faster the velocity changes direction; the larger the radius, the less rapidly the velocity changes direction. Since the acceleration is directed toward the center of the circle, the net force must be directed toward the center of the circle too. The net force must be applied by other objects.

object in circular motion

centripetal acceleration:

centripetal acceleration formula

where r is radius and v is velocity

centripetal force:

centripetal force formula

You can use your graphing calculator to determine how the magnitude of the centripetal force varies the speed with which the object is swung in the horizontal circle, the mass of the object, or the radius of the horizontal circle.

Graphing Calculator

Banking Angle

tan q = v2/rg

Kepler's Laws of Planetary Motion

  • Kepler's First Law: The path of each planet about the sun is an ellipse with the sun at one focus.
  • Kepler's Second Law: Each planet moves so that an imaginary line drawn from the sun to the planet sweeps out equal areas in equal periods of time.
  • Kepler's Third Law: The ratio of the squares of the periods (the time needed for one revolution about the sun) of any two planets revolving about the sun is equal to the ratio of the cubes of their mean distances from the sun.
    Kepler's Third Law

  • Motion in Two Dimensions Homework

    Periodic Motion Homework

    Motion in Two Dimensions Sample Problems