Optional Unit V: Applications of Kinematics and Dynamics
C. Projectile Motion

Key Concepts

In the absence of air resistance, all objects fall at a constant acceleration, regardless of their mass. This acceleration is called the acceleration due to gravity. It is given the symbol . It is a vector quantity. (Recent scientific discoveries suggest that mass may have a slight influence on the rate of acceleration of falling bodies.)

The magnitude of this acceleration is nearly 9.8 m/s² at the Earth's surface. The value changes slightly at different locations on the surface of the Earth.

The magnitude of is generally different on other celestial bodies.

Regardless of how an object is projected, it will experience a downward acceleration of nearly 9.8 m/s² near the surface of the Earth.

The value of can be determined experimentally.

One method of determining g involves measuring the period of a pendulum at low amplitude. (This is an approximate equation, valid when the angle of displacement, , (in radians) sin )

or,

When air resistance acts on falling bodies it may cause them to fall slower than they would in a vacuum.

Various factors affect the air resistance on a falling body.

In a medium such as air, falling bodies will eventually reach a terminal velocity.

The terminal velocity for objects has different values, depending on various factors such as mass, shape, size, and surface texture.

Some useful equations for free-fall motion are:

A projectile is any object which is thrown or otherwise projected into the air. (In this section, it is assumed that the effects due to air resistance are negligible for projectiles travelling at a low velocity relative to the air.)

The path that a projectile follows is called its trajectory.

A projectile that is projected (or fired) horizontally from some height follows a parabolic trajectory toward the ground. It takes the same time to reach the ground as if it were dropped from rest.

The shape of the trajectory of a projectile, as with any other type of object experiencing motion, is relative to the frame of reference of an observer. Observers in different frames of reference may describe a moving object differently.

To analyze projectile motion it is useful to consider the vertical (y) and horizontal (x) components of the velocity and displacement separately.

Horizontally, a projectile travels at a constant velocity.

Vertically, a projectile travels at the same rate as an object experiencing free-fall motion.

The horizontal motion of a projectile depends on the horizontal component of the initial velocity .

()

The horizontal displacement of a projectile, relative to the ground, can be found by using the horizontal component of the velocity and the total flight time (t) .

The vertical motion of a projectile depends on the vertical component of the initial velocity and on the acceleration due to gravity.

The vertical motion of a projectile can be determined with the equations for free-fall motion, using vertical components and the acceleration due to gravity.

A projectile fired from level ground at some firing angle (, measured relative to the ground,) other than 90° will follow a parabolic trajectory.

The maximum vertical displacement of a projectile fired from level ground will occur at one-half of the total time of flight. At its maximum vertical position the vertical component of the velocity vector is zero.

To obtain the maximum vertical displacement for a projectile with some constant magnitude for its initial velocity, the projectile should be released vertically, at a firing angle of 90°. It will also achieve the maximum time of flight in this way. (Equations 1 and 4 below give the maximum value for = 90°.

For a projectile fired from level ground, to achieve maximum horizontal displacement for some constant magnitude of the initial velocity, the projectile should be released at a firing angle of 45°. In situations where the weight of the projectile is similar to the applied force, firing angles less than 45° will produce the maximum horizontal displacement. (Equation 2 below gives the maximum value for = 45°).

If the magnitude of the initial velocity remains constant, projectiles fired from level ground at firing angles of (45° - ) and (45° + ), where 0< < 45° will have the same horizontal displacement. The one fired at (45° + ) will attain a higher maximum elevation and remain in flight longer than the one fired at (45° - ).

Basic projectile equations for an object fired at some initial velocity () at an angle from the horizontal over a horizontal surface are:

1) d_y = height =

2) d_x = range =

3) t_r = rise time = t_f = fall time =

4) flight time, t =

Learning Outcomes

Students will increase their abilities to:

1. Define the following terms: acceleration due to gravity, projectile, firing angle, trajectory, frame of reference, and terminal velocity.

2. Explain that mass will not substantially influence the motion of objects falling in a vacuum.

3. State the approximate value of the acceleration due to gravity for an object falling freely near the sur- face of the Earth.

4. State that an object dropped from rest experiences a downward acceleration.

5. Explain that an object thrown vertically upward experiences a downward acceleration.

6. Explain the effect of air resistance on falling bodies.

7. Predict the motion that would be experienced by various different types of falling objects.

8. Devise strategies for changing the terminal velocity of falling objects.

9. Transfer knowledge of motion in one or two dimensions to realistic situations.

10. Solve problems involving vertical free fall using equations for uniformly accelerated motion.

11. Recognize that projectile motion can be analyzed by considering the horizontal and vertical components of the motion separately .

12. Explain what factors affect the horizontal and vertical motion of a projectile.

13. Apply kinematic equations for constant velocity to analyze the horizontal motion of a projectile.

14. Apply kinematic equations for uniform acceleration to analyze the vertical motion of a projectile.

15. Solve a variety of problems related to projectile motion.

Teaching Suggestions, Activities and Demonstrations

1. Experimentally analyze the motion of an object experiencing free-fall.

2. Using a pendulum, determine the magnitude of the acceleration due to gravity near the surface of the Earth, with the relationship:

   

   or,

3. Observe the motion of different types of falling objects and account for any noticeable differences.

4. Perform an activity to compare the time it takes two objects to reach the ground, if one is released vertically, and another is simultaneously projected horizontally from the same height. Try to record the results on videotape for closer scrutiny afterwards.

5. Use stroboscopic photographs or illustrations to analyze the motion of two objects, one released vertically and the other projected horizontally from the same height at the same time.

6. Investigate the effects of mass, shape, or surface texture on the motion of falling objects.

7. Experimentally determine the trajectory of a projectile.

   Using a hose with a fine spray, allow a continuous stream of water to flow along a curved path. (Using a garden hose outside along the edge of a brick wall would allow chalk marks to be placed on the wall behind the spray as reference marks.) Analyze the curve. Plot the recorded points on a scaled graph and attempt to develop a relationship which gives the horizontal and vertical displacements of the curved path. If an appropriate computer software package is available, enter the data points corresponding to the vertical and horizontal displacements of the projectile path, and perform graphical analysis of the results.

8. As an alternative to the above procedure, place sheets of white paper and carbon paper on a large sheet of plywood, or a portable blackboard. Tilt the board at an angle. Roll a heavy steel ball up the ramp along a curved path. The carbon paper produces a permanent record of the track on the white paper for further analysis. The analysis can be done by measuring horizontal and vertical positions to obtain data for analysis.

   The results should yield a parabolic curve in the form: y = kx², because the vertical displacement d_yis based on:

   

9. Compare the maximum horizontal displacement, the maximum vertical displacement, and the total duration of flight for two projectiles having the same magnitude of their initial velocities but fired at (45°-) and (45°+), where 0< < 45°.

10. Push an air puck or a dry ice puck up a slight incline so that it follows a curved trajectory. Place markers along the path at equal time intervals. Analyze the path of the "projectile" in this simulation.

11. Using a recording timer and tape, drop a laboratory mass vertically from rest. Analyze the results. Tabulate data and draw displacement versus time and velocity versus time graphs. Use the velocity versus time graph to determine the acceleration of the falling mass. Repeat the experiment using a different mass. Perform similar experiments rolling carts down inclined planes. Develop a relationship between the slope of the inclined plane and the acceleration of the cart. Place additional mass on the cart and compare the rate of acceleration down the inclined plane for a heavy and light cart.

12. Use an air table and pucks, or dry ice pucks, to slide pucks down a slight incline. (Balloon-filled air pucks and plastic beads to lower friction are relatively inexpensive alternatives.) Use a metronome, a pendulum, or some other calibrated timer to determine the time needed for the puck to travel the total distance along the surface. As the puck travels, use markers to record the position of the puck at uniform time intervals. (Some fancier air tables include such things as spark timers or "blinkies.") Use the results to determine the acceleration of the puck.

13. Place a coin and a feather in a vacuum tube. Evacuate the tube with a vacuum pump. Invert the tube to show the relative rate at which a feather and a coin fall in the tube. Allow air to enter the tube. Invert again to illustrate the effect of air resistance on different objects .

14. Drop different spherical beads in a long glass column filled with a viscous liquid. Observe differences in the rate at which they drop. Compare when the same objects are released in air. Account for any differences in behaviour. This demonstration can also be used to illustrate the effects of buoyant forces on objects travelling through fluids.

15. Determine the vertical displacement an object will undergo in 0.20 second time intervals for 1.0 seconds. Measure out a piece of string that length and tie washers on the string at those displacements, starting from the bottom of the string. (If a location is available with a balcony or a safe overhang, you can use a longer length of string and washers representing longer time intervals.) Predict the sound that will be heard as the string falls. The washers should strike the ground at regular time intervals, even though they are not uniformly spaced. This illustrates concretely that as an object accelerates it undergoes increasing displacement with time over uniform time intervals.

16. For an interesting problem to explore, consider why the top section of a tree breaks off as the tree falls. A similar phenomenon occurs when a chimney falls. Certain sections along the solid object accelerate faster than if they were experiencing free fall.

    To illustrate this, tape a 100 mL plastic beaker to a meter stick. Place a ball in the beaker. Hold the meter stick down at one end, so it acts like a hinge as it falls. Allow the meter stick to fall, observing the ball carefully. Repeat, placing the beaker in a different location. (Better still, have several beaker-meter stick arrangements set up beforehand, to make the comparisons more quickly.) In a certain position, the beaker will drop faster than the ball. The ball will fall back into the beaker after the meter stick has attained a horizontal position. Repeat using balls of different mass to confirm that similar results occur regardless of the mass of the ball being used.

17. Using a special pendulum apparatus shown below, the value of g can be developed experimentally if the angle of displacement is small.

    

    When the thread is broken, the pendulum swings downward and the ball is released. Carbon paper on the arm of the pendulum marks the spot where the ball and pendulum arm make contact. Measuring the vertical displacement of the ball allows the time of travel to be determined. This time coincides with one-quarter of the period of the pendulum, assuming you already know the period of the pendulum.

    Since the sphere begins to fall from rest (v_i=0), its displacement is:
    , so
    where

18. Another apparatus which allows one to find g is the Kater pendulum. It has two weights, a heavy one and a lighter one, attached to a rigid beam in such a way that the distance separating the weights is adjustable. The pendulum can be swung from either end on a wall bracket, along two knife edges located just inside the position of the weights. When the period is the same with the pendulum swinging either from the top or bottom knife edge, the magnitude of g can be found by: ²l / T²

    where l is the separation between the two knife edges and T is the period of the pendulum.

    Instead of adjusting the position of the weights, the experiment can be conducted by adjusting one knife-edge. A plot of the period of the pendulum against knife-edge position will yield the magnitude of the acceleration due to gravity.

    The compound pendulum provides still another variation on the same theme for finding g. A pendulum attached to a spring support, so that it can oscillate laterally and vertically simultaneously, offers another approach. With some of these interesting pendulum variations available, different approaches increase student interest.

    Another variation is a variable "g" pendulum, where the axis of the angle of swing can be varied.

         

19. To show the effects of air resistance on falling objects, hold a piece of paper and a ball the same height above the ground. Drop the two simultaneously. Air resistance prevents the sheet of paper from falling at the same rate.

    Take the piece of paper and crumble it into a small ball. Again drop the paper and the ball simultaneously. This time the paper and ball fall at nearly the same rate, since the air resistance on the paper has been reduced.

    Another way to reduce air resistance on the sheet of paper is to place it on top of a book and then drop the two. The paper remains on top of the book as the two fall. This might also lead into a discussion of Bernoulli's Principle.

20. .A traditional game of corncob darts illustrates important principles in projectile motion and aerodynamics. Darts are made by attaching feathers to corncobs. These darts are then hurled underhand at a circular target on the ground. The size of the target and the distance away from the target can be changed, depending on the skill level of the participants.

    The object of the game is to place as many of your darts into the circle. Do not use metal-tipped darts for this activity. They could cause serious injury.

21. Various archery games can be used to illustrate principles of projectile motion and aerodynamics. Caution should be used whenever archery equipment is used.

22. Use computer simulations to analyze projectile motion.

23. Applications of projectile motion in hunting could be investigated.

24. Have students obtain various athletic implements (e.g. golf balls, footballs, soccer balls, shot putts, etc.). These can be thrown, kicked, or hit in an outdoor activity. By measuring the horizontal displacement and the time of travel, students may be able to calculate the initial velocities in both horizontal and vertical directions, as well as the initial firing angle.

25. Place a piece of paper under a book. Drop the two from waist height. Predict what would happen if the paper were on top of the book instead. Repeat. Account for the results.