Motion in Two Dimensions by: Heather Britton

Motion in Two Dimensions We have studied how motion takes place in terms of displacement, time, velocity, and acceleration in one dimension Now we will look at what happens in two dimensions using the same variables and equations

Motion in Two Dimensions Projectile motion - an object (projectile) moving freely in the air under the influence of gravity alone We will not concern ourselves with how the object was projected just what happens afterward

Motion in Two Dimensions Examples of projectiles include A thrown ball A speeding bullet A person jumping A skateboarder going off a curb

Motion in Two Dimensions Solving problems involving 2D motion involves working with vectors We will either add vectors to obtain a resultant Or we will resolve vectors into components

Motion in Two Dimensions When we work with components we will use the cartesian plane and x and y values This will allow us to form right triangles and use trigonometric identities Lets look at 3 different examples involving a flying plane

Motion in Two Dimensions Example 1 What is the resultant velocity (magnitude and direction) of a plane flying at 100 km/hr South with a tail wind of 20 km/hr?

Motion in Two Dimensions Example 2 What is the resultant velocity (magnitude and direction) of a plane flying at 100 km/hr South into a head wind of 20 km/hr

Motion in Two Dimensions Example 3 What is the resultant velocity (magnitude and direction) of a plane flying at 100 km/hr South and a cross wind blowing due East at 20 km/hr?

Motion in Two Dimensions http://www.physicsclassroom.com/mme dia/vectors/plane.cfm

Motion in Two Dimensions Example 4 You can swim with a velocity of 4 m/s. What would your resultant velocity (magnitude and direction) be if you were swimming across a river with a current of 1 m/s?

Motion in Two Dimensions When given a resultant velocity and an angle, you can find the component velocities (use x and y subscripts) Using the cartesian plane draw your resultant velocity and angle For projectile motion x is horizontal (adjacent) and y is vertical (opposite)

Motion in Two Dimensions Example 5 A swimmer crosses a river with a resultant velocity of 6 m/s. Because of the current she lands at an angle of 35 degrees from the horizontal. How fast does she swim, and how fast is the current?

Motion in Two Dimensions In projectile motion we can treat what happens in the x direction as a constant velocity problem We can treat what happens in the y direction as a constant acceleration problem www.physicsclassroom.com

Motion in Two Dimensions Practice finding component velocities (vox and voy) vo = 15 m/s θ = 0° vo = 8 m/s θ = 30° vo = 45 m/s θ = 62°

Motion in Two Dimensions http://www.physicsclassroom.com/mme dia/vectors/pap.cfm

Motion in Two Dimensions Since the velocity in the x direction is constant we can use the equation for constant velocity vx = x / t x represents the displacement in the x direction not the distance the projectile actually flew

Motion in Two Dimensions The y component velocity is undergoing constant acceleration due to gravity This means that vy is constantly changing The vy is changing at the rate of 9.8 m/s2 So with constant acceleration in projectile motion a = g = 9.8 m/s2

Motion in Two Dimensions The equations to be used for vy are 1. vy = voy + at 2. y = voyt + (1/2)at2 3. vy2 = voy2 + 2ay 4. avg vy = (vy + voy)/2

Motion in Two Dimensions The y component velocity will determine the time an object spends in the air The greater percentage of y component velocity the greater height the object will reach Whenever time needs to be determined use the y component velocity

Motion in Two Dimensions http://www.physicsclassroom.com/mme dia/vectors/nhlp.cfm http://www.physicsclassroom.com/Clas s/vectors/u3l2b.cfm

Motion in Two Dimensions Example 6 A cliff diver must clear the rocks on the shore. If the rocks jut out 5.0 m and the cliff is 60.0 m high, what is the minimum horizontal velocity the diver must leave the ledge with?

Motion in Two Dimensions Example 7 A cannon ball is launched at 40 degrees above the horizontal with an initial velocity of 50 m/s. (1.) What are the component velocities? (2.) How high does it go? (3.) How long is it in the air? And (4.) How far down range (x axis) does it land?

Motion in Two Dimensions Example 8 A cannon ball is launched at 50 degrees above the horizontal with an initial velocity of 50 m/s. (1.) What are the component velocities? (2.) How high does it go? (3.) How long is it in the air? And (4.) How far down range (x axis) does it land?

Motion in Two Dimensions Examples 7 and 8 show the significance of complementary angles Complementary angles add up to 90° For complementary angles the range (horizontal displacement) will be the same The height attained and the time spent in the air will be different

Motion in Two Dimensions There is one angle that will give the greatest range That angle is 45° Therefore maximum horizontal displacement can be obtained by launching a projectile at a 45° angle

Motion in Two Dimensions Example 9 On a 130 m par 3 a ball is hit off the tee. 5.2 s later it lands in the hole. What was the initial velocity and what angle was the ball hit with?