Volume 1, Physics 208

The Science of Motion: Kinematics and Newton's Laws

Last updated on March 17, 1999

Introduction.

The science of motion is called kinematics. In order for us to systematically study the motions of objects on earth, we must learn the concepts of displacement, velocity and acceleration. The flight of a pitched or struck baseball, the long jump, the rise and dunk of a basketball player are all described by this science.

At first, we will describe these motions as they would be on earth if there were no effects of air (resistance or lift). Since sports are much influenced by air, we will go beyond the simplest vacuum physics and apply the effects of the air. In baseball, the air effects are what causes some of the most interesting effects of the curve ball, the rising fast ball and the knuckle ball.

The physics of why things move the way they do is the science of dynamics. We will learn this after the kinematics.

Velocity.

The average velocity of an object is determined by the displacement divided by the time interval. Displacement has a magnitude and a direction. Displacement is a vector.

Dx is a vector in the x direction. Dx= x2 -x1, where x1 and x2 are the beginning and end points. [Bold type signifies a vector. In class, I will use a little arrow on top of the character.]

Dt = t2-t1.

v = Dx/Dt where we will use the International units for our calculations (meters/second)

1 m/s = 2.3 mi/hr

In 100 m races Carl Lewis would usually come from behind to win the race in the last 20 m. Measurements have been made of the racer's velocities vs. time. The distance-time and velocity-time curves of Lewis vs. the others demonstrate how the races are run by the various racers. The measured velocity curves show that Lewis reaches top speed later but holds his high speed. Others seem to reach top speed earlier but are unable to hold their speed and actually slow down in the latter half of the race.

The addition of velocities.

If you are swimming with a velocity of 2 m/s directly across a river is your velocity affected by the current?

This depends on your point of view. Take two observers.

  1. A person in an inner tube floating on the river
  2. A person standing on the shore.

Imagine what you would see from each point of view.

  1. For the person in the inner tube floating on the river [and moving with the current] the swimmer always has a velocity of 2m/s for all velocities of the river's current. For the swimmer aimed across the river, the swimmer would take 50 seconds to cross a 100 m wide river and would appear to always be heading directly across the river.
  2. For the observer on shore, the swimmer's velocity is the vector sum of the current's velocity + the swimmer's velocity [as measured in the water]. The observer will see that the swimmer has taken 50 seconds to cross but the swimmer will also be carried downriver by the current for 50 seconds. If the current were flowing at 4 m/s, the swimmer will have been carried 200 m downstream.

The simpler description is obtained by the observer in the inner tube! The swimmer always has the same velocity Independent of the velocity of the current of the river. This is an important point for our future consideration. Whether you take the position of observer 1 or 2 depends on the question being asked.

If you want to know the swimmer's performance, then the inner tube would be the right place from which to observe.

Accelerated Motion.

It used to be said of some athletes, that they are slow but they are quick. [Steve Largent, the great pass receiver was so described.] What does this mean? Does it make sense?

Sports writers usually mean that a quick athlete is one who can change his direction or his speed very quickly. This makes the athlete very elusive. Though he couldn't blow by a defender with sheer speed, a defensive back trying to cover Largent would find himself left in the dust as Largent would make his quick moves to get free to catch a pass.

In the language of physics we would say that the athlete has great acceleration. An athlete with great acceleration can change direction or speed quickly. In our new language, the change of either direction or speed is just the change in velocity.

Definition. Acceleration , A = Dv/Dt

Acceleration is the slope of the velocity-time curve.

Falling bodies.

An object falling to the floor from a height of one meter takes about 0.5 seconds. It starts with zero velocity and ends up with velocity pointing downwards. Is this accelerated motion? What is the average velocity for this journey?

Examine a displacement-time curve for a falling body. Convert this to a velocity-time curve and then to an acceleration time curve.

Newton's falling Apple

The pictures and analysis come from the package "World in Motion"

The red spot marks the apple center at time intervals of 1/30 second.

The displacement-time data. The horizontal axis (abcissa) is the time and the vertical axis (ordinate) is the displacement.

From direct measurements of displacement vs. time we can derive the following:

The velocity vs. time obtained from the displacement vs. time data. v = Dy/Dt. Note that the velocity is changing linearly (almost) with time. Since v = At, the slope of the v-t curve is the acceleration. The slope is negative, so A is a negative (points downward). The acceleration comes from the earth's gravity. In outer space, a released ball would not accelerate.

The acceleration vs. time data derived from the A = Dv/Dt. The value of A is about -10m/s2.

When a major league pitcher pitches, the ball takes about 0.5 seconds to reach the plate. During this time, does the ball fall toward the ground? Explain.

A major league pitch. The red dots are spaced apart by 1/30second. The angle of the camera distorts the image. It's 60' or 18 m from the mound to the plate. [The ball is travelling about 1.5 m/click as did the hand just before delivery.] There are about 13 clicks from release until the ball is struck. vavg = 18 m / (13/30) s = 41.5 m/s = 95.5 m/hr.

In the graphs below, don't pay any attention to the values on the axes; they aren't calibrated. Look at the shape of the curves of the vertical displacement, vertical velocity, and vertical acceleration.

Displacement vs time: Between the times of the pitcher's release and the batters hit, the y vs. t graph gives a nice parabola just like the ball tossed upwards and continually accelerated downwards.

Velocity vs. time: The velocity downward increases linearly just like the dropped ball with constant acceleration. Note that vy starts out positive; the ball is aimed slightly upwards...even for a 95 mph fast ball!!



Acceleration vs time: The acceleration remains constant just like in the dropped ball.

We have studied the description of motion in terms of displacement, velocity and acceleration and how these are related to one another. Now the question we will tackle is why accelerations occur. Intuitively we comprehend that we must apply some motive force to accelerate objects or ourselves. However, the force hypothesis is sometimes counter-intuitive. Newton discovered the laws relating acceleration and force. By so doing, he discovered the laws governing the perpetual motion of the planets as well as the more prosaic motions of the objects on Earth.

Newton's Laws of Motion.

1. An object will remain at rest or with constant speed in straight line motion unless a force is impressed upon it.

2. F = mA where F and A are vectors. F is the net force; the vector sum of all the forces.

Comment.

If there are several forces acting on a point, you must find the vector sum of the forces. It is the net force which compels the mass to undergo acceleration. If you impress two equal forces which are in opposite directions on a point mass, the mass will have no acceleration.

3. For every action there is an equal and opposite reaction.

Running or jumping depends on thrusting against the ground. It is the reaction force of the ground on your body that accelerates you for sprinting or vertical jumping.

At first glance it might seem that if action and reaction are equal, then it's impossible to have a net force and hence an acceleration. The point is that the action and reaction act on different bodies. When you push against the ground, the ground is accelerated away from you (though not very much, of course); the ground pushes on you to push you into the air. The forces are equal and opposite, but they do not both act on the same object.

Units of Force and Mass.

In the International System of units, we have

Force: newtons (N). 1 Newton = 0.22 pounds

Mass: kilogram (kg). (We don't have a common unit of mass in British units)

Some example calculations with only one force.

A mass of 2 kg rests on a horizontal frictionless table. A force of 10 N pushes on this mass. Find the acceleration.

F = mA, A = F / m = 10 / 2 = 5 m/s2 in the same direction as the force.

An object with a net force of 25 N pulling on it is observed to have an acceleration of 8 m/s2. What is the mass of the object?

m = F / A = 25 / 8 kg = 3.1 kg

Multiple Forces: The Free Body Diagram.

If there are multiple forces acting on an object you must resolve them into a single vector some---or net force---before applying F = mA . A useful tool to help accomplish this is the free-body diagram. A free-body diagram is basically just a book-keeping tool to help you keep track of all of the various forces. In a free-body diagram, the object being acted on is represented by a point, whether it is an atom or a galaxy. Then the external forces are drawn as vectors originating at the point. This makes it easy to add up all of the vectors, component by component, and arrive at the net force. Let's try a simple example:

Consider a ball with a mass of 4 kg attached to a string. You pull upwards on the string with a force of 100 N. What is the acceleration of the ball?

First, make a free body diagram:
Now find the net force; if we take up to be positive then we have 100 N - 40 N = 60 N. Since this is positive we know that the net force is up. Now apply Fnet = mA : A = F / m = 60 / 4 = 15 m/s2

Questions.

  1. If you were capable of throwing a pitch at 40 m/s and you threw the ball while on a train going 50 m/s. What velocity would a ground observer measure for the pitched ball? Give the answer for 3 cases: ball thrown the same direction as the train's velocity, the opposite direction, and transverse to the train's velocity.


  2. If you threw the ball at 40 m/s from a pitcher's mound to the plate, a distance of 18 m, how long would the ball take to travel to the plate?


  3. You are buzzing a baseball stadium in your jet fighter. Flying at 100 m/s in the direction from the pitcher's mound to homeplate, your radar detects a pitched ball to have a velocity of 130 m/s. How long will it take for the pitched ball to go from the mound to the plate?


  4. A swimmer wanting to safely cross a rather wide river must decide how to aim himself in his swim. The river is flowing with a rather high velocity but the water is smooth and not turbulent at all. How should he aim himself to make the passage? Should he choose the strategy of swimming the shortest distance? the shortest time? or in between these? Let the river be 100 m wide with a current flowing at 5 m/. The swimmer can swim [in still water] 2 m/sec. What is the shortest time for the journey and how far does he go [as measured by an land observer] and how long does it take? What is the shortest possible distance for the journey and how long does it take?


  5. From the description of Carl Lewis' 100 m dash, would you say that he had great acceleration? Great speed?. Explain your answer in terms of the performance curve [velocity-time curve].

  6. If Lewis takes 4 seconds to reach his top speed of 11 m/s, what is his average acceleration?

  7. Express the following lengths in meters: a) Bob Beaman's long jump (29' 2.5") b) Bob Seagren's pole vault (18' 5.75") c) Jay Silvester's discus throw (224' 5").


  8. Express in seconds: a) 8 m + 13.8 s b) 2 hr,10 m c) 1 week


  9. Convert 55 MPH to a) km/hr b) m/s.


  10. A woman finishes a marathon in 2 hr and 20 m. The marathon distance is 26 mi + 384 yards. Find her average velocity in m/s.


  11. Determine the velocity per one second interval and the overall average velocity for an object moving in accordance with the following data: 
         x (m)              time (s)                       
     
       0                 0         

      2.7                1         

      21.6               2         

      72.9               3         

     172.8               4

  12. Use the data below to obtain the position at t = 0.5 s and at 1.5 s. Plot the data from this table and make a smooth curve through the points. Find the instantaneous velocity at 0.5 and 1.5 seconds. 
        time(s)            x(m)        

       0                 0         

       1                 1         

       2                 4         

       3                 9
  13. Construct a graph of position versus time having the property that: a) the velocity is always negative; b) the velocity is never zero; c) the velocity is always positive


  14. The quarter-mile splits for a miler are 0:59, 1:57, 3:03, and 4:02 (minutes:seconds). What is the average velocity for the entire race and for each of the 1/4 miles?


  15. Runner A completes 1500 m at an average velocity of 7.3 m/s while runner B requires less time at an average velocity of 7.4 m/s. Determine the time for each runner and the time interval between the finishes.


  16. Use the velocities in question 15. If runner A starts 3 seconds before runner B how far from the start line will B catch up to A? How long after A starts will this happen? Make a displacement-time graph to prove the solution to this problem.


  17. A pitch travels 17 m from the pitcher to the plate. Calculate the time required to reach the plate if a) the pitch has a velocity of 100 mi/hr b) the pitch has a velocity of 80 mi/hr.


  18. A small (1000 kg) car is sitting on a frictionless surface. You exert a force of 500 N on the car. What is the acceleration of the car? If you apply a rather small force like 2 N, can you move the car? With what acceleration?


  19. A skateboarder initially cruising at 14 m/s falls off the board onto the roadway and comes to a halt in 5 m. What is the acceleration (magnitude and direction) of the skater? What is the force for a 70 kg skater? Where does the force come from?


  20. A cyclist is pushing backwards against the ground (via the wheels) with a force of 25 N while going at a constant velocity. What is the net force on the cyclist? What are all the forces acting on the cyclist? Make a free body diagram and name the forces.


Conversion of units.

Answers to questions.

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