Understanding Projectile Motion: Precision and Accuracy

1. This power point has animations that will be lost in a print-out.For best understanding, view power point in slide show mode.

2. Warm-up – Shoot a dart at the bullseye! BULLSEYE

3. Nonlinear Motion Projectile Motion

4. Precision and Accuracy Experimental results can be characterized by their precision and their accuracy. Precision describes the degree of exactness of a measurement. The precision of a measurement is one-half the smallest division of the instrument (ruler, graduated cylinder, etc.) A meterstick’s smallest division is the millimeter, so you can measure the length of an object to within half a millimeter. Accuracy describes how well the results of an experiment agree with the standard value. Instruments may need to be calibrated before a precise measurement will be accurate. correct result correct result correct result correct result precise but not accurate accurate but not precise not accurate or precise accurate and precise

5. Projectile Motion • Projectile motion is nonlinear motion – motion along a curved path. • The object in projectile motion has two independent components of motion: • horizontal motion • vertical motion

6. Projectile Motion Projectile motion problems are best solved by treating horizontal and vertical motion separately. Note – gravity only affects vertical motion.

7. Vector and Scalar Quantities (Review) • A vector quantity as both magnitude and direction. • A scalar quantity has only magnitude. • Velocity is a vector, as is acceleration. • Scalars include quantities that can be specified with only magnitude such as mass, volume, time, etc.

8. Velocity Vectors • An arrow is used to represent the magnitude and direction of a vector quantity. • A velocity is sometimes the result of combining two or more other velocities. • For example, an airplane’s velocity is a combination of the velocity of the airplane relative to the air and the velocity of the air relative to the ground, or the wind velocity. • Consider the airplane show here. The airplane is flying north at 100 km/h relative to the surrounding air. • With a tailwind of 20 km/h, the plane is flying at a velocity of 120 km/h relative to the ground. • With a headwind of 20 km/h, the place is flying at a velocity of 80 km/h to the ground.

9. 100 km/h 80 km/h resultant 60 km/h Velocity Vectors (Review) • Consider the airplane show above. The airplane is flying north at 80 km/h relative to the surrounding air. • With a crosswind of 60 km/h, the plane is flying at a velocity of 100 km/h relative to the ground. • This resultant was found using the parallelogram method and the Pythagorean Theorem.

10. Components of Vectors • A velocity vector can be resolved into an equivalent set of two component vectors at right angles to each other – the horizontal component and the vertical component. • The resolution of a vector is just the parallelogram method done backwards.

11. Surfing nicely illustrates component and resultant vectors. When surfing in the same direction as the wave, our velocity is the same as the wave’s velocity, v. This velocity is called v because we are moving perpendicular to the wave front. To go faster, we surf at an angle to the wave front. Now we have a component of velocity parallel to the wave front, vll, as well as the perpendicular component v. We can vary vll, but v stays relatively constant as long as we ride the wave. Adding components, we see that when surfing at an angle to the wave front our resultant velocity, vr, exceeds v. As we increase our angle relative to the wave front, the resultant velocity also increases. Physics of Sports - Surfing

12. Projectile Motion • Projectiles near the surface of the earth follow a curved path that can be resolved into horizontal and vertical components. • The horizontal component of motion for a projectile is just like the horizontal motion of a ball rolling freely along a level surface. Neglecting friction, the rolling ball moves at constant velocity. • The vertical component of a projectile’s velocity is like the motion for a freely falling object. In the vertical direction, the projectile accelerates downward due to gravity. • The horizontal component of motion for a projectile is completely independent of the vertical component of motion. Their combined effects produce the variety of the curved paths of projectiles.

13. Projectile Motion A dropped object and a projectile will hit the ground at the same time because gravity is the only force affecting the vertical vector.

14. Two General Types of Projectile Motion Objects launched horizontally Objects launched upwards at an angle

15. Consider the cannonball shot at an upward angle in the picture. Because of gravity, the cannonball follows the curved path as shown. If there were no gravity, the cannonball would follow a straight-line path such as shown by the dashed line. The vertical distance the cannonball falls at any point beneath this imaginary dashed line is the same vertical distance it would fall if it were dropped from rest and had been falling the same amount of time. Recall that this distance is given by d = ½gt2, where t is the elapsed time. Rounding g to 10 m/s2, at one second the cannonball is 5 m below the dashed line; at 2 seconds it is 20 m below; at 3 s its 45 m below, etc. Upwardly Launched Projectiles

16. Note that the cannonball moves equal horizontal distances in equal time intervals. This is because there is no horizontal acceleration, the only acceleration is due to gravity in the vertical direction. Upwardly Launched Projectiles

17. The angle and initial velocity the projectile is launched will determine the distance the projectile will travel – the horizontal range. The picture shows a soccer ball launched at the same initial speed but at different angles. Upwardly Launched Projectiles • Notice that: • the soccer ball reaches different heights, • the paths are all parabolas, • the 45° path has the longest horizontal range, and • any two paths whose angles add up to 90° will have the same horizontal range (75° and 15° paths, 30° and 60° paths, etc).

18. All the previous examples were with negligible air resistance. In the presence of air resistance, the path of a high speed projectile falls below the idealized parabola and follows a solid curve. If air resistance is negligible, a projectile will rise to its maximum height in the same time it takes to fall from that height to the ground. This is due to the constant effect of gravity. The deceleration due to gravity going up is the same as the acceleration due to gravity coming down. The projectile will hit the ground with the same speed it had when it was projected upward. Upwardly Launched Projectiles

19. Objects Launched at an Angle vy v h = maximum height vx v = initial velocity Rx = horizontal range θ = launch angle t = total time in air v = initial velocity horizontal vx = v cos θ Rx = vxt vertical vy = v sin θ h = vyt/4 t = 2vy/g q = launch angle

20. Calculations t = t/2 = time in air to highest point h = maximum height t = total time in air θ = launch angle v = initial velocity Rx = horizontal range g = 9.8 m/s2 horizontal vx = v cosθ Rx = vxt vertical vy = v sin θ From average acceleration equations: vy = gt; g = vy/t; solve for h. h = ½ gt2 = ½(vy/ t)t2 = ½vyt = vyt/4 = h h = ½ gt2 = ½ vyt ; t = 2vy/g

21. Important Facts The horizontal velocity is constant. It rises and falls in equal time intervals. It reaches maximum height in half the total time. Only gravity effects the vertical motion.

22. Objects Launched Horizontally Rx = horizontal range vx = initial horizontal velocity t = total time in the air vx h = height above ground horizontal vertical Rx = vxt h = ½gt2

23. Important Facts There is no horizontal acceleration. There is no initial vertical velocity. The horizontal velocity is constant. Time is the same for both vertical and horizontal.

24. Check Question The boy on the tower throws a ball a distance of 20 m. At what speed is the ball thrown? • The ball is thrown horizontally, so its speed equals the horizontal distance divided by the time: v = d/t or vx = Rx/t • We also know that h = ½gt2. Since h = 5 m, t must equal 1 s. • Solving out, we get v = d/t = 20 m / 1 s = 20 m/s

25. For short range projectile motion such as a batted ball or a cannonball, we usually assume the ground is flat. However, for very long range projectiles the curvature of Earth’s surface must be taken into account. If an object is projected fast enough, it will fall around the Earth and become an Earth satellite. An Earth satellite, such as the space shuttle or the moon, is simply a projectile traveling fast enough to fall around Earth rather than into it. At the speed necessary to fall around Earth, 8 km/s, most objects would burn up in the atmosphere. This is why satellites are launched at altitudes above 150 km – high enough not to burn up but still affected by gravity. Fast-Moving Projectiles – Satellites

26. Nonlinear Motion Uniform Circular Motion

27. Uniform Circular Motion • Can an object be accelerated if its speed remains constant? • Yes, if the change in velocity is a change in the object’s direction, not its speed. • Uniform circular motion is the movement of an object or point mass at constant speed around a circle with a fixed axis. • Example: a rider on a merry-go-round is in uniform circular motion.

28. Circular MotionRevolution and Rotation Revolution – object moves in circular path around an external point. Ex: Revolution around an external point: the Earth revolves around the sun. Rotation – object moves in a circular path around an internal point or axis. Ex: Rotation around an axis: the Earth rotates (or spins) on its axis.

29. Revolutionaround an external point An object revolves at constant speed. The path is a perfect circle

30. The radius for circular motion is a vector (red arrow). This radius vector locates the orbiting object. One should imagine an x, y coordinate system with its origin at the center of the circle. The radius extends from this origin to the position of the object. The velocity vector (green arrow) shows the speed and direction of the orbiting object at all points along its path. Note that the velocity vector is tangent to the circular path of the object and is perpendicular to the radius vector at all points on the orbit. Vectors of Circular Motion

31. Centripetal Force According to Newton’sFirst Law of Motion, an object moves in a straight line unless a force acts on it to make it turn. An external force is necessary to make an object follow a circular path. This force is called a centripetal (“center seeking”)force.

32. Centripetal Acceleration Since everyunbalanced force causes an object to accelerate in the direction of that force(Newton’s Second Law – F = ma), a centripetal force causes a centripetal acceleration. This acceleration results from a change in direction, anddoes not imply a change in speed,although speed may also change. In uniform circular motion the speed does not change and the centripetal acceleration results only from the change in position. The centripetal acceleration vector of the object always points in toward the center of the circle (center-seeking).

33. Centripetal force and acceleration may be caused by: friction – car rounding a curve As a car makes a turn, the force of friction acting upon the turned wheels of the car provide the centripetal force required for circular motion. a rope/cord – swinging a mass on a string As a bucket of water is tied to a string and spun in a circle, the force of tension acting upon the bucket provides the centripetal force required for circular motion. gravity – planets orbiting the sun As the Earth orbits the sun, or as the moon orbits the Earth, the force of gravity acting upon the moon provides the centripetal force required for circular motion. Examples:

34. Period In all cases of uniform circular motion, a mass m moves in a circular path of radius r with a linear (tangential) speed v. The time to make one complete revolution is known as the period, T. v r m The speed v is the circumference divided by the period. v = 2pr T

35. Formulas: centripetal acceleration (m/s2) ac= v2/r and centripetal force (N) Fc=mac =mv2/r m = mass in kg v = linear velocity in m/s r = radius of curvature in m

36. Formulas Using Period: centripetal acceleration (m/s2) ac=v2= 4π2r r T2 and centripetal force (N) Fc=mac=m 4π2r T2 m = mass in kg v = linear velocity in m/s r = radius of curvature in m

37. The Fictitious Force Centrifugal Force is a fictitious force which is actually the absence of a centripetal force. It’s called fictitious because centrifugal forces exists only in rotating reference frames, not in inertial (constant velocity) reference frames.

38. Why Rotate a Space Station ? Centrifugal force is a fictitious force that occurs in a rotating system. This 'force' can be used to simulate gravity in space where there is no solid surface to enable us to feel the forces of gravity. The centrifugal force is in a direction perpendicular to the rotation axis and radially outward. As a result the astronauts in the space station are able to walk around inside the space station as if the artificial gravity is pulling them outward away from the center of the donut shaped station.

39. Rotationaround an axis Rotational motion - object moves in a circular path about an internal point or axis (“rotates” or “spins”)

40. Angular Displacement The amount (distance) that an object rotates is its angular displacement. Angular displacement, θ, is given in degrees, radians, or rotations. 1 rotation = 360 deg = 2π radians θ

41. Degrees and Radians There are 360 in a circle. Another common unit of angle measure (particularly in circles) is radians. There are 180 in radians and 360 in 2 radians. Radians are useful when dealing with calculations involving revolutions. 2 radians = 1 complete revolution. When working with radians, it is customary to work with fractions of .

42. Tangential Speed Recall that linear speed is the distance moved per unit time. In circular motion, this term can be used interchangeably with the term tangential speed. Tangential speed is the speed of an object moving in a circular path.

43. Avoid Confusion Do NOT confuse tangential speedwithangular speed or rotational speed, which is the number of rotations per unit time. Angular speed (or velocity), ω, is given in deg/s, rad/s, rpm, etc...

44. Example: If two ladybugs sit on a rotating object at different distances from the axis, they will each have the same rotational speed but different tangential speeds.

45. Angular Acceleration An object’s angular acceleration, α, is given in deg/s2, rad/s2, rpm/s, etc... Formulas for rotational motion follow an exact parallel with linear motion formulas. The only difference is a change in variables and a slight change in their meanings.

46. Constant Acceleration Formulas Constant LINEAR ROTATIONAL wf= wi + at vf = vi + at q = wavt d = vavt vav =(wf+ wi)/2 vav = (vf + vi)/2 q = wit+½ at2 d = vit + ½ at2 vf2 = vi2 + 2ad wf2= wi2+2aq

47. Rotational-Linear Parallels

48. P E R O D I C MOTION

49. Periodic motion - any motion in which the path of the object repeats itself in equal time intervals. Thesimple pendulumis a great example of this type of motion.

50. The period, T, of a simple pendulum (time needed for one complete cycle) is approximated by the equation: where l is the length of the pendulum and g is the acceleration of gravity.