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2D Motion. Chapter 3. Introduction. In order to talk about movement in 2 dimensions, we have to talk about vectors. What is a vector? A vector is a quantity with both a magnitude and a direction.
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2D Motion Chapter 3
Introduction • In order to talk about movement in 2 dimensions, we have to talk about vectors. • What is a vector? • A vector is a quantity with both a magnitude and a direction. • When drawing a vector, the arrow points in the direction of the motion and is proportional to the magnitude.
Adding Vectors • When adding scalars, you can add them arithmetically. However, vectors have that pesky direction to deal with. • For instance, what if you travel 3 miles west and then 2 miles east?
Adding Vectors • Ultimately, you walked 1 mile to the west. • This is your resultant or net vector. • You can add the vectors graphically, as long as they lay in the same plane.
Adding Vectors • What if they aren’t in the same plane? • What if you go North 20 meters and East 35 meters? • In order to add these two vectors together, you can use the tail to tip method.
Subtracting Vectors • What if you are asked to subtract a vector? • Subtracting a vector is the same as adding a vector going in the opposite direction.
Components • Imagine that someone gave you directions to their house by telling you to head 37 miles 23o north of east. How would you do that? • You could get a compass or you can break the vector into its components.
Trig Review R y q x • When using vectors, you will need to use trigonometry. y = R sin q x = R cos q R2 = x2 + y2
Problem 1 300 90 m • Find the height of a building if it casts a shadow 90 m long and the indicated angle is 30o.
Problem 2 • A person walks 400 m in a direction of 30o N of E. How far is the displacement east and how far north?
Problem 3 • Find the components of the 240-N force exerted by the boy on the girl if his arm makes an angle of 280 with the ground.
Resultants • When you have the components of the vector, you can also get the resultant. • When you calculate the resultant, you will have to get the direction as well.
Identifying Direction N 60o 50o W E 60o 60o S A common way of identifying direction is by reference to East, North, West, and South. Length = 40 m 40 m, 50o N of E 40 m, 60o N of W 40 m, 60o W of S 40 m, 60o S of E
Problem 4 • A 30-lb southward force and a 40-lb eastward force act on a donkey at the same time. What is the net or resultant force on the donkey?
Multiple Vectors • If you have more than two vectors, you can still find the resultant by adding up all the individual components.
Problem 5 • A boat moves 2.0 km east then 4.0 km north, then 3.0 km west, and finally 2.0 km south. Find resultant displacement.
Projectiles • What is a projectile? • A projectile is any object that is thrown into the air that is subject to gravity. • Can you think of some examples?
Projectile motion is simplified by the use of vectors. • Imagine a baseball is hit and is now traveling through the air. • The ball has both a horizontal movement and vertical movement.
Projectile motion can also be called free fall with a horizontal velocity.
The ideal path of a projectile is a parabola. • Why don’t projectiles travel in a true parabola?
Free Fall with a Horizontal Velocity • Projectiles can be classified as free fall problems with a horizontal velocity. • Consider Bob – he drops a ball off of a 220 m building. What can we figure out? • Now what if Bob throws the ball off of the same building but he throws it east with a velocity of 3.5 m/s. What can we figure out?
The Range • The only difference between those two setups is that the second ball lands a specific distance away from the building. The first ball ends up at the base of the building. • This specific distance is called the range.
The Equations • Consider this equation: x=xo + vot + 1/2at2 Let’s examine the horizontal direction. What is the acceleration in the horizontal direction? So what happens to the equation?
The Equations • Consider this equation: x=xo + vot + 1/2at2 Let’s examine the vertical direction. What is the initial velocity in the vertical direction? So what happens to the equation?
X vs. Y • For a zero degree angle, horizontal direction is now x=vot • For a zero degree angle, vertical direction is now y=1/2at2
Of Importance… • The vertical component is completely independent of the horizontal component and vice versa. • Remember that gravity is negative. • The horizontal velocity is constant.
Problem 6 • Bob is standing on top of a bridge 45.0 m high. He kicks a rock off the bridge with a horizontal velocity of 5.5 m/s. How far does the rock travel?
So far we have only dealt with projectiles that are launched at a an angle of zero degrees. What is the path of the motion? • What if we took the that projectile and shot it up at an angle of 10o? • Whenever a projectile is shot at an angle, you have to deal with the horizontal and vertical components of its resultant path.
Projectiles at an Angle • When a projectile is launched at an angle, it will have an initial vertical velocity as well as its initial horizontal velocity. • The vertical component of the projectile is going to be vosin Θ . • The horizontal component of the projectile is going to be vocos Θ .
Projectile Table • Now let’s put all of this in a table:
The Formulas • Since v is constant throughout, the formula to find the direction in the horizontal direction is the same: x = vot • Just remember to substitute vx= vocos Θ for v. x = (vocos Θ)t
The Other Formula • The vertical component is a little bit trickier. • With a projectile at 0o, there was no initial vertical velocity so the second term of the equation cancelled out. Now there is an initial velocity so: y = (vosin Θ) t + 1/2gt2
Problem • A foot ball is kicked at an angle of 37o with a velocity of 20.0 m/s. Calculate the following: • Maximum height • Time to travel before the football hits the ground • How far away it hits the ground • The velocity vector at the maximum height • The acceleration vector at the maximum height
