Slide 1 / 91 Slide 2 / 91 AP Physics C - Mechanics Kinematics in Two Dimensions 2015-12-03 www.njctl.org Slide 3 / 91 Slide 4 / 91 Table of Contents Click on the topic to go to that section Vector Notation · Projectile Motion · Vector Notation Uniform Circular Motion · Relative Motion · Return to Table of Contents Slide 5 / 91 Slide 6 / 91 Position and Velocity Vectors Motion problems in one dimension are interesting, but frequently, objects are moving in two, and even three dimensions (four, when you count time as a dimension in special and general relativity). This is where the vector notation learned earlier comes in very handy, and we will start by defining a position vector, .
Slide 7 / 91 Slide 8 / 91 Average Velocity As an object moves from one point in space to another, the average velocity of its motion can be described as the displacement of the object divided by the time it takes to move. (average velocity vector) Slide 9 / 91 Slide 10 / 91 Instantaneous Velocity Instantaneous Velocity Components The instantaneous velocity has three different components: v x , To find the instantaneous velocity (the velocity at a specific point in v y , and v z (any of which can equal zero). time) requires the time interval to be so small that it can effectively be reduced to 0, which is represented as a limit. Each component is shown below: (instantaneous velocity vector) Vector representation: Notation note: it will be assumed that all the motion vectors are time dependent, so after this slide x(t), y(t) and z(t) will be shown as x, y and z (same convention for velocity and acceleration). Slide 11 / 91 Slide 12 / 91 Average Acceleration Instantaneous Acceleration Acceleration is the rate at which the velocity is changing, and the Just as we can find the velocity at a specific point in time, we can average acceleration can be found by taking the difference of the also find the instantaneous acceleration using a limit. final and initial velocity and dividing it by the time it takes for that event to occur.
Slide 13 / 91 Slide 14 / 91 1 The vector, , Instantaneous Acceleration describes the position of a particle as a function of time. Find the expression for the velocity and acceleration The instantaneous acceleration has three different components: vectors expressed as a function of time. a x , a y , and a z (any of which can equal zero). Each component is shown below: Vector representation: Slide 15 / 91 Slide 16 / 91 2 The vector, , describes Integration the position of a particle as a function of time. Find the expression for the velocity and acceleration vectors The unit on One Dimension Kinematics showed how to obtain expressed as a function of time. position from velocity, and velocity from acceleration through integration techniques. The same method works for two and three dimensions. Each component is shown below, and since we are only looking for instantaneous values, we will leave out the limits of integration: Slide 17 / 91 Slide 18 / 91 3 The vector, , describes the Integration acceleration of a particle as a function of time. Find the expression for the velocity and position vectors Here is it what it looks like from a vector point of view, where we expressed as a function of time. start with acceleration and integrate twice to get to position:
Slide 19 / 91 Slide 20 / 91 Instantaneous values Once the vector for position, velocity or acceleration is found, either by differentiation or integration, the instantaneous value can be found by substituting the value of time in for t. Notation note: When you find the value of the position, velocity or vector, just leave it in vector notation - don't worry about the units - at this point in your physics education, its assumed you know them! Slide 21 / 91 Slide 22 / 91 6 What is the velocity of an object at t = 3 s if its acceleration is described by ? Slide 23 / 91 Slide 24 / 91 Projectile Motion Have you ever thrown an object in the air or kicked a soccer ball to a friend and watched the path in space it followed? The path is described by mathematics and physics - it is a parabolic path - another reason why you studied parabolas in mathematics. Projectile Motion v y v x v x v x v y v v y v x v x v y The above is an x-y plot that shows the path of the object - and shows at Take a minute and various points, the velocity vectors. discuss the behavior Return to Table of the v y vectors. of Contents
Slide 25 / 91 Slide 26 / 91 Projectile Motion Projectile Motion The v y vectors are acting as studied earlier - v y is maximum at Just as in mathematics where a vector is resolved into two the launch point, decreases under the influence of the perpendicular vectors (x and y), in real life, the x motion is gravitational field, reaches zero at the apex, and then independent of the y motion and can be dealt with separately. increases until it reaches the negative of the initial velocity v y v x right before it strikes the ground. v x v x v y v v y v y v x v x v x v x v y v v x v y v y v x The v y vectors change because after launch, the only force v x acting on the ball in the y direction is gravity. But, neglecting v y friction, there are NO forces acting in the x direction. Now that the v y behavior has been reviewed, what else do you notice about this picture? So v x is constant throughout the motion. Slide 27 / 91 Slide 28 / 91 Projectile Velocity Velocity of a Projectile v y v x v y v x v x v x v y v x v v y v x v y v v y v y v x v x v x v x Vector analysis for the velocity gives us: v y v y In 1D Kinematics, you are used to the velocity of the object at its v total apex being zero. For 2D Kinematics, the y velocity is zero, but v y v y it has a total velocity because it still has a velocity component in the x direction. θ v x What is the direction of the acceleration vector at each point? Slide 29 / 91 Slide 30 / 91 Acceleration of a Projectile Motion of a Projectile v y v x v x v x v y v v y a y = -g v x a y = -g v x a y = -g v y You know from experience that this motion is a parabola. Let's see if this can be derived a y = -g a y = -g mathematically, by examining the position equations in the Near the surface of the planet Earth, there is zero acceleration x and y direction. in the x direction, and a constant acceleration, with magnitude, g, in the negative y direction. This is true, regardless of the direction of the velocity or displacement of the projectile. In the absence of a given initial point, we are free to set x 0 = y 0 = 0. The acceleration in the x direction is zero, a x = 0 a y = -g and the acceleration in the y direction is "-g."
Slide 31 / 91 Slide 32 / 91 Motion of a Projectile Motion of a Projectile v y v x v y v x v x v x v x v y v v y v x v y v v y v x v x v x We're using parametric equations here, v y v x where t is the parameter, and we're free to v y manipulate the x and y equations Here's the plot - looks simultaneously, since they both are true for like the sketch above. any given t. For more info - see your math That's good. teacher! The constants are combined, represented by A and B, and y is now expressed in terms of x. Plug A = 2.8 and B = 0.18 and see what your graphing calculator or other electronic device plots. Slide 33 / 91 Slide 34 / 91 7 Which of the following statements are true regarding Projectile Displacement v y projectile motion? v x v x v x v y v v y A is constant. v x v x v y B Total acceleration is +g when the object is rising We now have a group of equations and vectors to work with to predict the future motion of a projectile, given initial conditions. and -g when it is falling. v total C In the absence of friction, the trajectory depends v y v y upon the mass of the object. θ v x D The velocity of the object is zero at the apex of the motion. E The horizontal motion is independent of the vertical motion. Slide 35 / 91 Slide 36 / 91 8 A marble is shot by a marble launcher and follows a 9 A marble is shot by a marble launcher and follows a parabolic path as shown. Air resistance is negligible. parabolic path as shown. Air resistance is negligible. Which of the following shows the direction of the velocity Which of the following shows the direction of the net at point Y, the highest point on the path? force at point X? A None. A None. B B C C D D E E
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