Projectiles, Circular & Harmonic Motions

Summary and Practice of Analysis

© by William Dietsch 1998


• The motion of objects, which move through curved paths, is the result of the body having components to its velocity, which are two dimensional in nature.  The two dimensions of motion are independently analyzed and combined to describe the actual path of the body.

• Independence of the motions is the key to the understanding and simple mathematical analysis of curved motion.

• Symmetry of events is often a useful tool in the analysis of this sort of motion.  Here the symmetry is that the path of a body moving through a regular two-dimensional path is the same at the beginning and end of the event.

Projectile motion:

• Galileo and others, in observing the motions of falling objects from moving platforms, concluded that projectiles and objects dropped from moving platforms have exactly the same characteristics of their paths.

• An object, which is launched and proceeds through space freely falling only under the influence of gravity, is called a projectile.  Golf balls, shot arrows, things dropped from moving aircraft, artillery shells, and bullets are all considered to be projectiles.  Objects under power such as rockets or airplanes are not projectiles.

• Projectile motion is classified and dealt with in this unit as either launched horizontally or launched upward at an angle.

• The path through which a projectile moves is called its trajectory.  The shape of the trajectory of a projectile (neglecting air friction) is a parabola.  A projectile displays symmetry in its trajectory if the effects of friction are ignored.

Circular motion:

• As the name implies, the motion of a body moving in a circular path is called circular motion.  A body does not need to move through a complete circle in order to undergo circular motion.  Motion through an arc with a fixed radius also constitutes circular motion.

• A body can move through a horizontal circle at constant speed.  This is called uniform circular motion and the body undergoes constant acceleration, even though the speed remains constant.  The direction of the velocity vector constantly changes and thus the body is accelerating.

• The acceleration of a body undergoing circular motion is called the centripetal acceleration (ac).  Equations: ac = v2/r (where v is the instantaneous, but constant speed, preferably in m/s, and r is the radius of the circle) and ac = 4π2r/T2 (where T is the period of one revolution).

• The force, which causes centripetal acceleration, is called the centripetal force (Fc). The centripetal force is the force, acting toward the center of the circle, responsible for causing the centripetal acceleration.  Equations: Fc = mac (where m is the mass of the body) or Fc = m v2/r and Fc = m4π2r/T2.  Fighter pilots making high-speed turns experience large centripetal forces which can sometimes cause them to lose consciousness.  The centripetal force divided by g (9.8 m/s2) is referred to as g forces.  Humans in good condition can tolerate up to 9 g's without physical harm (except for a brief loss of consciousness).

Harmonic motion:

Simple harmonic motion is linear motion in which the acceleration is proportional to the distance from the equilibrium position and is directed toward that position.  Examples of harmonic motion are a swinging pendulum and a weight bobbing up and down on a spring.

Periodic motion occurs when a body repeatedly moves over the same path in equal intervals of time.

• A pendulum is an object suspended so that it can swing back and forth about an axis.  The motion of a pendulum is independent of its mass.  The motion of a pendulum is independent of its amplitude (in swings of 100 or less?).  The period of a pendulum is directly proportional to the square root of its length.  The period of a pendulum is inversely proportional to the square root of the local acceleration of gravity (and so is sometimes used to measure local gravity).  The pendulum equation:

Frequency is the number of cycles that an object in periodic motion will complete in a set period of time (preferably seconds).  The unit for frequency (ν) is Hertz (Hz), which stands for cycles per second.  The reciprocal of frequency is the period (T):  T = 1/ν.  The period is the number of seconds it takes to complete one complete cycle of periodic motion.

Practice problems for projectile motion launched horizontally

  1. Divers at Acapulco dive from a cliff that is 61 m high. If the rocks below the cliff extend outward for 23 m, what is the minimum horizontal velocity a diver must have to clear the rocks safely?

  2. A dart player throws a dart horizontally at a speed of 12.4 m/s.  The dart hits the board 0.32 m below the height from which it was thrown. How far away is the player from the board?

  3. A plane flying horizontally at 350 m/s releases a package at an altitude of 1500-m.
    1. How long will the package take to reach the ground?
    2. How far will it move horizontally while falling?

  4. In a TV tube, electrons are projected at a speed of 9.5 x 105 m/s toward the screen 0.45-m. away.  If gravity were the only force acting, how far would the electron fall when it hits the screen?

Projectiles launched upward at some angle

  1. An arrow is shot at a 30.0° angle with the horizontal. It has a velocity of 49 m/s.
    1. How high will the arrow go?
    2. What horizontal distance will it travel?

  2. In setting up a chase scene for a movie, a stunt coordinator builds a 45° take off ramp.  If a car is to be driven off the ramp at a speed of 50.0 m/s, how far away should the landing ramp be placed?

  3. A pitched ball is hit by a batter at a 45° angle. It just clears the outfield fence, 98-m away.  Find the velocity of the ball when it left the bat. Assume the fence is the same height as the pitch.

  4. A basketball player tries to make a half court jump shot, releasing the ball at the height of the basket.   Assuming the ball is launched at 51.0°, 14.0 m from the basket, what velocity must the player give the ball?

  5. A baseball is hit at 30.0 m/s at an angle of 53° with the horizontal.  Immediately an outfielder runs 4.00 m/s toward the infield and catches the ball at the same height it was hit.  What was the original distance between the batter and the outfielder?

uniform horizontal circular motion

  1. An athlete whirls a 7.00kg hammer tied to the end of a 1.3m chain in a horizontal circle.  The hammer moves at the rate of 1.0 RPS.
    1. What is the centripetal acceleration of the hammer?
    2. What is the tension in the chain?

  2. Sue whirls a yo-yo in a horizontal circle. The yo-yo has a mass of 0.20 kg and is attached to a string 0.80 m long.
    1. If the yo-yo makes 1.0 complete revolution each second, what force does the string exert on it?
    2. If Sue increases the speed of the yo-yo to 2.0 revolutions per second, what force does the string now exert?

  3. A 5.0-g coin is placed on a stereo record revolving at 33 1/3 RPM.
    1. In what direction is the acceleration of the coin, if any?
    2. Find the acceleration of the coin when it is placed 5.0, 10, and 15 cm from the center of the record.  What force accelerates the coin?

  4. According to the Guinness Book of World Records, (1990 edition, p. 169) the highest rotary speed ever attained was 2010 m/s (4500 mph).  The rotating rod was 15.3 cm (6 in) long. Assume the speed quoted is that of the end of the rod.
    1. What is the centripetal acceleration of the end of the rod?
    2. If you were to attach a 1.00g object to the end of the rod, what force would be needed to hold it on the rod?
    3. What is the period of rotation of the rod?

  5. A ride called the rotor has a 2.0m radius and rotates 1.1 RPS.
    1. Find the speed of a rider on the rim of the rotor.
    2. Find the centripetal acceleration of the rider.

  6. It takes a 615kg racing car 14.3 s to travel at a uniform speed around a circular racetrack of 50.0-m radius.
    1. What is the acceleration of the car?
    2. What average force must the track exert on the tires to produce this acceleration?

Pendulum motion

  1. A pendulum makes 35 complete oscillations in 12 s.
    1. What is its period?
    2. What is its frequency?

  2. A pendulum is 3.500 m long.  What is its period at the North Pole?  (Where the acceleration of gravity is 9.832 m/s2).  What is its period in Java?  (where the acceleration of gravity is 9.782 m/s2)

  3. A pendulum has a frequency of 5.50 Hz on earth at a point where g = 9.80 m/s2.  What would be its frequency on Jupiter? (where the acceleration of gravity is 2.54 times that on earth)

  4. A pendulum extends from the roof of a building almost to the floor.  If the pendulum's period is 8.5 s, how tall is the building?


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created and © 1998 by William Dietsch
posted & edited 7 April 2007 by D Trapp
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