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Chapter 13 – Vector Functions 13.4 Motion in Space: Velocity and Acceleration Objectives: Determine how to calculate velocity and acceleration. Determine the motion of an object using the Tangent and Normal vectors. 13.4 Motion in Space: Velocity and Acceleration 1 Position Vector Suppose a particle moves through space so that its position vector at time t is r(t). Notice from the figure that, for small values of h, the vector approximates the direction of the particle moving along the curve r(t). Its magnitude measures the size of the displacement vector per unit time. 13.4 Motion in Space: Velocity and Acceleration 2 Velocity Vector The vector 1 gives the average velocity over a time interval of length h and its limit is the velocity vector v(t) at time t : The velocity vector is also the tangent vector and points in the direction of the tangent line. 13.4 Motion in Space: Velocity and Acceleration 3 Speed The speed of the particle at time t is the magnitude of the velocity vector, that is, |v(t)|. | v ( t ) | | r '( t ) | ds = rate of change w .r.t. tim e dt For one dimensional motion, the acceleration of the particle is defined as the derivative of the velocity: a(t) = v’(t) = r”(t) 13.4 Motion in Space: Velocity and Acceleration 4 Visualization Velocity and Acceleration Vectors 13.4 Motion in Space: Velocity and Acceleration 5 Example 1 Find the velocity, acceleration, and speed of a particle with the given position function. r ( t ) t i ln t j t k 2 13.4 Motion in Space: Velocity and Acceleration 6 Newton’s Second Law of Motion If the force that acts on a particle is known, then the acceleration can be found from Newton’s Second Law of Motion. The vector version of this law states that if, any any time t, a force F(t) acts on an object of mass m producing an acceleration a(t), then F(t) = ma(t) 13.4 Motion in Space: Velocity and Acceleration 7 Example 2 – pg. 871 # 28 A batter hits a baseball 3 ft above the ground toward the center field fence, which is 10 ft high and 400 ft from home plate. The ball leaves the bat with speed 115 ft/s at an angle of 50o above the horizontal. Is it a home run? (Does the ball clear the fence?) 13.4 Motion in Space: Velocity and Acceleration 8 Tangential and Normal Components of Acceleration When we study the motion of a particle, it is often useful to resolve the acceleration into two components: ◦ Tangential (in the direction of the tangent) ◦ Normal (in the direction of the normal) a v 'T v N 2 13.4 Motion in Space: Velocity and Acceleration 9 Tangential and Normal Components of Acceleration Writing aT and aN for the tangential and normal components of acceleration, we have a = aTT + aNN where aT = v’ and aN = v2 13.4 Motion in Space: Velocity and Acceleration 10 Tangential and Normal Components of Acceleration We will need to have aT = v’ and aN = v2 in terms of r, r’, and r”. To obtain these formulas below, we start with v · a. aT v ' r '( t ) r ''( t ) aN v 2 r '( t ) r '( t ) r ''( t ) r '( t ) 13.4 Motion in Space: Velocity and Acceleration 11 Example 3 – pg. 871 # 38 Find the tangential and normal components of the acceleration vector. r ( t ) 1 t i t 2 t j 2 13.4 Motion in Space: Velocity and Acceleration 12 Kepler’s Laws Note: Read pages 844 – 846. 1. A planet revolves around the sun in an elliptical orbit with the sun at one focus. 2. The line joining the sun to a planet sweeps out equal areas in equal times. 3. The square of the period of revolution of a planet is proportional to the cube of the length of the major axis of orbit. 13.4 Motion in Space: Velocity and Acceleration 13 More Examples The video examples below are from section 13.4 in your textbook. Please watch them on your own time for extra instruction. Each video is about 2 minutes in length. ◦ Example 3 ◦ Example 5 ◦ Example 6 13.4 Motion in Space: Velocity and Acceleration 14 Demonstrations Feel free to explore these demonstrations below. Kinematics of a Moving Point Ballistic Trajectories 13.4 Motion in Space: Velocity and Acceleration 15