Kinematics. Vector solutions. Vectors

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1 Kinematics Study of motion Accelerated vs unaccelerated motion Translational vs Rotational motion Vector solutions required for problems of 2- directional motion Vector solutions Possible solution sets Solve problems graphically Treat vectors as triangles Break vectors into x- and y- components Treat all vectors as i- j- problem All solution methods are acceptable. Vectors Problem While flying a in plane, the plane s velocity was 150 m/s at 50 North of West At the same time, the wind was blowing 30 m/s at 20 East of North. Diagram the problem Solve as a triangle Break into x- and y- components Express in i, j vector format

2 Notations Variable Name Standard Notation AS/A2 Text Notation Displacement d, x, or y x or s Velocity v, v f v Initial Velocity v i or v o u Acceleration a a Subscripts In most American texts, you ll see initial or final indicated by a subscript. The subscript indicating initial is usually an i or a 0. In AS/A2, u is used to mean initial velocity. Numeric subscripts such as v 1 and v 2 may also be used New Terms Objects that move without rotation are said to be in TRANSLATIONAL motion. Objects moving are usually said to be moving in REFERENCE to something else. A fly traveling in a car would be moving in reference to the car and in reference to an outside observer.

3 Coordinate Systems In all problems of motion and force, we must start by establishing a coordinate system and a reference frame. A reference frame is the view point of our observer of the problem HOMEWORK QUESTION: DEFINE INERTIAL REFERENCE POINT The coordinate system is a point of origin and a selection of positive and negative direction. Time and Displacement Time is a scalar concept. It is an arbitrary measurement; it doesn t really measure a quantity itself, it measures changes in other quantities. The standard unit for time is the second. Displacement is a vector quantity. It is the magnitude and direction of position measured from a point. Distance is a scalar concept. Both are measured in meters Velocity Velocity is a measurement of the rate and direction of change in displacement. v avg = d/ t Instantaneous velocity is the velocity at a infinitely small time interval. We can approximate instantaneous velocity by making our time interval increasingly small or by calculating a tangent line.

4 Relative Velocity A boat is trying to cross a river that is 750 m wide. The boat s velocity is 5 m/s relative to the water. After crossing the river, the boat finds itself 50 meters further downstream than it was initially. Determine the displacement of the boat. Determine the velocity of boat relative to the shore. Determine the velocity of the current. Relative Velocity and other vectors in i,j notation Complete the same problems with i,j notation.. Acceleration Acceleration is the measurement of the rate and direction of change in velocity. a avg = v/ t Instantaneous acceleration is the acceleration at a infinitely small time interval. We can approximate instantaneous acceleration by making our time interval increasingly small or by calculating a tangent line. The change in acceleration over time is called JERK

5 Kinematics Formulas!! v f = v i + at d f = d i + v i t + ½at² d f = d i + ½(v f + v i )t v f ² = v i ² + 2a(d f - d i ) Gravitational Acceleration Close to the surface of the Earth, we consider gravitational acceleration to be a constant. g = 9.8 m/s 2 Sometimes this acceleration is referred to as 1 g. 2g s would be 19.6 m/s 2 Gravitational Acceleration Any falling object accelerates vertically downward at a rate of 9.8 m/s 2 regardless of any acceleration in the horizontal direction. Ignoring air resistance, an object would continue to gain velocity downward. In the real world, objects will eventually achieve terminal velocity.

6 Ignoring Air Resistance Any object in free-fall experiences acceleration in the y-axis at a rate of What is the y-velocity of an object at the peak of its flight? More Free-Fall Tips The time it takes for an object to travel to the peak of its flight is the same as the time it takes to travel back down to its starting position! The velocity at the peak is zero m/s The acceleration up and down is the same its 9.8 m/s² downward ALWAYS In groups A runner hopes to complete the 10,000-m run in less than 30.0 min. After exactly 27.0 min, there are still 1,100 m to go. The runner must then acceleration at 0.20 m/s² for how many seconds in order to achieve the desired time. Solve the problem. Graph a distance-time graph, a velocity-time graph and an acceleration-time graph for the problem.

7 Reminder: Displacement Time Graphs A displacement time graph always shows time on the x-axis and displacement (travel or position) on the y-axis. The gradient of this line is VELOCITY If the line is straight, the velocity is constant If the line is curved, there is acceleration Reminder: Velocity Time Graphs A velocity time graph shows time on the x-axis and velocity on the y-axis The gradient of the velocity-time graph is acceleration If the line is straight, the acceleration is constant If the line is curved, there is JERK The area under a Velocity-Time graph is DISPLACEMENT Q21

8 Q22 Make it Linear! Graphs of displacement time with acceleration are not linear Describe the shape of the graph given that d= ½at² Making this linear means showing t² versus d OR t versus sqrt(d) The slope can then be related to acceleration HOW? #49 a) When is the velocity the greatest? b) During what intervals, if any, is the velocity constant? c) During what intervals, if any, is the acceleration constant? d) When is the MAGNITUDE of the acceleration greatest?

9 #51 a) At what time is the velocity greatest? b) At what time, if any, is the velocity zero? c) Does the object move in one direction or both directions during the time shown? Problem #55 MAKE THE VELOCITY TIME GRAPH TO GO WITH THE POSITION TIME GRAPH! Problem #52 & 53 #52 a) Estimate Acceleration in 2 nd and 4 th gear b) Estimate how far the car traveled in 4 th gear #53 a) Estimate the average acceleration during i) 1 st gear, ii) 3 rd gear, and iii) 5 th gear. b) What is the average acceleration during the first 4 gears. c) Estimate the distance traveled in 1 st gear.

10 Questions about the test? Sketch graphs of d-t, v-t and a-t with and without air resistance Use motion equations and graphs to solve kinematics problems. Add vectors and use i,j format Relative velocity questions Cambridge formula notations Projectile Motion Projectile motion is the term used to define an object moving in both the x and y directions Remember, the acceleration of freely falling bodies is always 9.8 m/s 2 downward! Projectile Motion An airplane was 1250 meters from the ground traveling at 200 m/s, How long would it take an object thrown from the plane to drop to the ground? How far did the object travel horizontally in that time? Does the speed of the airplane change the time the object takes to hit the ground?

11 Standard Notation Displacement in the horizontal (x) direction = d x Displacement in the vertical (y) direction = d y Velocity in the horizontal (x) direction = v x Velocity in the vertical (y) direction = v y Acceleration in the horizontal (x) direction = a x Acceleration in the vertical (y) direction = a y. Velocity Given the initial angle of an object undergoing projectile motion ( ), you can determine the vector components of the velocity by: v x = v cos( ) v y = v sin( ) Range Equations Pages in the AS/A2 text cover the development of the range formula for projectile motion. Please note, this only works for objects that end at the same vertical point as they start.

12 Range Formula d f = d i + (v i sin )t ½ gt 2 since d f = d i 0 = (v i sin )t ½ gt 2 We can solve this equation for t as t = 0 or t = (v i sin ) /g Deriving the Range Formula Part 1. Develop a range formula for objects with an initial horizontal velocity. Part 2. Develop a range formula for objects with an angled initial velocity Range Formula Substituting this into the equation for RANGE (d f in the x direction) d f = (v i cos )t (no acceleration!) d f = (v i cos )(v i sin ) /g d f = v i2 (sin )(cos ) /g d f = v i2 (sin2 )/g

13 Period and Frequency Objects moving in circular motion can be defined by their period and their frequency. Period (t) is the amount of time an object takes to complete 1 revolution and is measured in seconds. Frequency (f) is the revolutions per unit of time and is measured in Hertz (Hz). The formula relating frequency and period is: f = 1/t Tangential Velocity Objects moving in a circular path cover a distance equal to the circumference of the path times the number of revolutions. So for one revolution, we can write the formula for velocity as v = (2 r)/t Centripetal Acceleration Centripetal means towards the center ; centripetal acceleration is the acceleration acting on the object from the circular motion. The formula for centripetal acceleration is: a c = v²/r

14 Total Acceleration In addition to having centripetal acceleration, an object may have TANGENTIAL acceleration (if the tangential velocity is not constant). The total acceleration could then be expressed as the vector sum of the centripetal acceleration and the tangential acceleration a total = (a c ² + a t ²) where the angle = tan-1(at/ac) Radians In angular velocity and angular acceleration, we express motion in terms of radians. Any angle can be expressed in terms of radians by radians = /2. One complete revolution is 2 radians (6.28 radians) or 360. Angular Velocity Angular velocity is the same for all points and is dependent on frequency or period of the motion. = (2 )/t = 2 f = rad /t From angular velocity, you can calculate tangential velocity of a point on the radius v = r

15 Acceleration Centripetal and Angular We can express centripetal acceleration in terms of angular velocity by a c = ²r Angular acceleration is angular velocity over time and occurs when the object is not in a state of constant angular velocity. = a t /r Motion Formulas for Rotation The formulas can be summarized as = 0 + t = 0 t + ½ t ² ² = 0 ² + 2 Challenge In an action-adventure film the hero is supposed to throw a grenade from his car, which is going 60 km/hr to his enemy s car which is going 110 km/hr. The enemy s car is 14.6 m in front of the hero s when he lets go of the grenade. If the hero throws the grenade so that its initial velocity relative to the hero is at an angle of 45 above the horizontal, what should be the magnitude of the velocity? The cars are both traveling in the same direction on a level road.

16 Definitions 1. Define displacement, velocity and acceleration; give appropriate units for each 1-d Kinematics Calculate problems of 1-d motion under constant velocity or constant acceleration. You will need all the kinematics equations; know how to interpret buried information such as dropped, peak velocity, etc Describe the d-t, v-t and a-t graphs for 1-d motion. Relate the graphs to slopes and area under the curve Air Resistance/Drag Relate free-fall without air resistance to free-fall WITH air resistance. Describe how air resistance changes acceleration and velocity.

17 Linear Graphs Relate graphical data of distance and time to velocity and acceleration. Special note: if an object falls from an initial velocity of 0 m/s; the displacement is equal to ½gt². How could this be graphed LINEARLY? 2-d Motion (Projectiles) Calculate problems of projectile motion for any initial angle of velocity. Remember to break angled velocity into constant x velocity = vcosθ and accelerated y velocity = vsinθ Diagram problems of projectile motion indicating velocity and acceleration vectors at any point along the path Circular Motion Convert between degrees and radians 360 = 2 radians Determine tangential velocity and centripetal acceleration for circular motion; give directions for each Determine velocity, displacement and acceleration using angular components rather than linear. = angular acceleration in rad/s² θ = angular displacement in radians ω = angular velocity in rad/s

18 General Knowledge Describe and define random and systematic error Define and describe accuracy and precision Discuss possible methods to improve labs Determine percentage error for values given actual and measured data