1. Newton’s Laws of Motion Claude A Pruneau Physics and Astronomy Wayne State University PHY 5200 Mechanical Phenomena Click to edit Master title style Click to edit Master subtitle style 1 Projectile Motion PHY 5200 Mechanical Phenomena Claude A Pruneau Physics and Astronomy Department Wayne State University Dec 2005.

2. Content Projectile Motion –Air Resistance –Linear Air Resistance –Trajectory and Range in a Linear Medium –Quadratic Air Resistance Charge Particle Motion –Motion of a Charged Particle in a Uniform Field –Complex Exponentials –Motion in a Magnetic Field

3. Description of Motion with F=ma F=ma, as a law of Nature applies to a very wide range of problems whose solution vary greatly depending on the type of force involved. Forces can be categorized as being “fundamental” or “effective” forces. Forces can also be categorized according to the degree of difficulty inherent in solving the 2nd order differential equation F = m a. –Function of position only –Function of speed, or velocity –Separable and non-separable forces In this Chapter –Separable forces which depend on position and velocity. –Non separable forces.

4. Air Resistance Air Resistance is neglected in introductory treatment of projectile motion. Air Resistance is however often non-negligible and must be accounted for to properly describe the trajectories of projectiles. –While the effect of air resistance may be very small in some cases, it can be rather important and complicated e.g. motion of a golf ball. One also need a way/technique to determine whether air resistance is important in any given situation.

5. Air Resistance - Basic Facts Air resistance is known under different names –Drag –Retardation Force –Resistive Force Basic Facts and Characteristics –Not a fundamental force… –Friction force resulting from different atomic phenomena –Depends on the velocity relative to the embedding fluid. –Direction of the force opposite to the velocity (typically). True for spherical objects, a good and sufficient approximation for many other objects. Not a good approximation for motion of a wing (airplane) - additional force involved called “lift”. –Here, we will only consider cases where the force is anti- parallel to the velocity - no sideways force.

6. Air Resistance - Drag Force Consider retardation force strictly anti- parallel to the velocity. Where   f(v) is the magnitude of the force. Measurements reveal f(v) is complicated - especially near the speed of sound… At low speed, one can write as a good approximation:

7. Air Resistance - Definitions Viscous drag Proportional to viscosity of the medium and linear size of object. Inertial Must accelerate mass of air which is in constant collision. Proportional to density of the medium and cross section of object. For a spherical projectile (e.g. canon ball, baseball, drop of rain): Where D is the diameter of the sphere  and  depend on the nature of the medium At STP in air:

8. Air Resistance - Linear or Quadratic Often, either of the linear or quadratic terms can be neglected. To determine whether this happens in a specific problem, consider Example: Baseball and Liquid Drops A baseball has a diameter of D = 7 cm, and travel at speed of order v=5 m/s. A drop of rain has D = 1 mm and v=0.6 m/s Millikan Oil Drop Experiments, D=1.5 mm and v=5x10 -5 m/s. Neither term can be neglected.

9. Air Resistance - Reynolds Number The linear term drag is proportional to the viscosity,  The quadratic term is related to the density of the fluid, . One finds Reynolds Number

10. Case 1: Linear Air Resistance Consider the motion of projectile for which one can neglect the quadratic drag term. From the 2nd law of Newton: Independent of position, thus: Furthermore, it is separable in coordinates (x,y,z). By contrast, for f(v)~v 2, one gets coupled y vs x motion A 1 st order differential equation y x Two separate differential equations Uncoupled.

11. Case 1: Linear Air Resistance - Horizontal Motion Consider an object moving horizontally in a resistive linear medium. Assume v x = v x0, x = 0 at t = 0. Assume the only relevant force is the drag force. Obviously, the object will slow down Define (for convenience): Thus, one must solve: Clearly: Which can be re-written: with Velocity exhibits exponential decay

12. Case 1: Linear Air Resistance - Horizontal Motion (cont’d) Position vs Time, integrate One gets

13. Vertical Motion with Linear Drag Consider motion of an object thrown vertically downward and subject to gravity and linear air resistance. Gravity accelerates the object down, the speed increases until the point when the retardation force becomes equal in magnitude to gravity. One then has terminal speed. y x Note dependence on mass and linear drag coefficient b. Implies terminal speed is different for different objects.

14. Equation of vertical motion for linear drag The equation of vertical motion is determined by Given the definition of the terminal speed, One can write instead Or in terms of differentials Separate variables Change variable: where

15. Equation of vertical motion for linear drag (cont’d) So we have … Integrate Or… Remember So, we get Now apply initial conditions: when t = 0, v y = v y0 This implies The velocity as a function of time is thus given by with

16. Equation of vertical motion for linear drag (cont’d) We found At t=0, one has Whereas for As the simplest case, consider v y0 =0, I.e. dropping an object from rest.

17. Equation of vertical motion for linear drag (cont’d) Vertical position vs time obtained by integration! Given The integration yields Assuming an initial position y=y 0, and initial velocity v y = v y0. One gets The position is thus given by y x

18. Equation of vertical motion for linear drag (cont’d) Note that it may be convenient to reverse the direction of the y-axis. Assuming the object is initially thrown upward, the position may thus be written y x

19. Equation of motion for linear drag (cont’d) Combine horizontal and vertical equations to get the trajectory of a projectile. To obtain an equation of the form y=y(x), solve the 1st equation for t, and substitute in the second equation.

20. Example: Projectile Motion x (m) y (m) No friction Linear friction

21. Horizontal Range In the absence of friction (vacuum), one has The range in vacuum is therefore For a system with linear drag, one has A transcendental equation - cannot be solved analytically

22. Horizontal Range (cont’d) If the the retardation force is very weak… So, consider a Taylor expansion of the logarithm in Let We get Neglect orders beyond We now get This leads to

23. Quadratic Air Resistance For macroscopic projectiles, it is usually a better approximation to consider the drag force is quadratic Newton’s Law is thus Although this is a first order equation, it is NOT separable in x,y,z components of the velocity.

24. Horizontal Motion with Quadratic Drag We have to solve Rearrange Integration Yields Solving for v Note: for t= , Separation of v and t variables permits independent integration on both sides of the equality… where at t = 0. with

25. Horizontal Motion with Quadratic Drag (cont’d) Horizontal position vs time obtained by integration … Never stops increasing By contrast to the “linear” case. Which saturates… Why? ! ? The retardation force becomes quite weak as soon as v<1. In realistic treatment, one must include both the linear and quadratic terms.

26. Measuring the vertical position, y, down. Terminal velocity achieved for For the baseball of our earlier example, this yields ~ 35 m/s or 80 miles/hour Rewrite in terms of the terminal velocity Solve by separation of variables Integration yields Solve for v Integrate to find Vertical Motion with Quadratic Drag

27. Quadratic Draw with V/H motion Equation of motion With y vertically upward

28. Motion of a Charge in Uniform Magnetic Field Another “simple” application of Newton’s 2nd law… Motion of a charged particle, q, in a uniform magnetic field, B, pointing in the z-direction. The force is The equation of motion The 2nd reduces to a first order Eq. Components of velocity and field Z x y

29. Motion of a Charge in Uniform Magnetic Field (cont’d) Three components of the Eq of motion Define Rewrite Cyclotron frequency Coupled Equations Solution in the complex plane …

30. Complex Plane O x (real part) y (imaginary part) Representation of the velocity vector

31. Why and How using complex numbers for this? Velocity Acceleration Remember Eqs of motion We can write Or

32. Why and How using … (cont’d) Equation of motion Solution Verify by substitution

33. Complex Exponentials Taylor Expansion of Exponential The series converges for any value of z (real or complex, large or small). It satisfies And is indeed a general solution for So we were justified in assuming  is a solution of the Eqs of motion.

34. Complex Exponentials (cont’d) The exponential of a purely imaginary number is Separation of the real and imaginary parts - since i 2 =-1, i 3 =-I We get Euler’s Formula where  is a real number

35. Complex Exponentials (cont’d) Euler’s Formula implies e i  lies on a unit circle. O x y 1

36. Complex Exponentials (cont’d) A complex number expressed in the polar form O x y a where a and  are real numbers Amplitude Phase Angular Frequency

37. Solution for a charge in uniform B field v z constant implies The motion in the x-y plane best represented by introduction of complex number. The derivative of  Integration of  Greek letter “xi”

38. Solution for a charge in uniform B field (cont’d) Redefine the z-axis so it passes through (X,Y) which for t = 0, implies Motion frequency O x y

39. Solution for a charge in uniform B field (cont’d) O x y Helix Motion