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Vectors and Direction • In drawing a vector as an arrow you must choose a scale. • If you walk five meters east, your displacement can be represented by a 5 cm arrow pointing to the east. Vectors and Direction • Suppose you walk 5 meters east, turn, go 8 meters north, then turn and go 3 meters west. • Your position is now 8 meters north and 2 meters east of where you started. • The diagonal vector that connects the starting position with the final position is called the resultant. Vectors and Direction • The resultant is the sum of two or more vectors added together. • You could have walked a shorter distance by going 2 m east and 8 m north, and still ended up in the same place. • The resultant shows the most direct line between the starting position and the final position. Calculate a resultant vector • An ant walks 2 meters West, 3 meters North, and 6 meters East. • What is the displacement of the ant? 7.1 Finding Vector Components Graphically • Draw a displacement vector as an arrow of appropriate length at the specified angle. • Mark the angle and use a ruler to draw the arrow. 7.1 Finding the Magnitude of a Vector • When you know the x- and y- components of a vector, and the vectors form a right triangle, you can find the magnitude using the Pythagorean theorem. 7.1 Adding Vectors • Writing vectors in components make it easy to add them. 7.1 Subtracting Vectors 7.1 Calculate vector magnitude • A mail-delivery robot needs to get from where it is to the mail bin on the map. • Find a sequence of two displacement vectors that will allow the robot to avoid hitting the desk in the middle. 7.2 Projectile Motion and the Velocity Vector • Any object that is moving through the air affected only by gravity is called a projectile. • The path a projectile follows is called its trajectory. 7.2 Projectile Motion and the Velocity Vector • The trajectory of a thrown basketball follows a special type of archshaped curve called a parabola. • The distance a projectile travels horizontally is called its range. 7.2 Projectile Motion and the Velocity Vector • The velocity vector (v) is a way to precisely describe the speed and direction of motion. • There are two ways to represent velocity. • Both tell how fast and in what direction the ball travels. 7.2 Calculate magnitude Draw the velocity vector v = (5, 5) m/sec and calculate the magnitude of the velocity (the speed), using the Pythagorean theorem. 7.2 Components of the Velocity Vector • Suppose a car is driving 20 meters per second. • The direction of the vector is 127 degrees. • The polar representation of the velocity is v = (20 m/sec, 127°). 7.2 Calculate velocity • A soccer ball is kicked at a speed of 10 m/s and an angle of 30 degrees. • Find the horizontal and vertical components of the ball’s initial velocity. 7.2 Adding Velocity Components • Sometimes the total velocity of an object is a combination of velocities. One example is the motion of a boat on a river. The boat moves with a certain velocity relative to the water. The water is also moving with another velocity relative to the land. 7.2 Adding Velocity Components 7.2 Calculate velocity components • An airplane is moving at a velocity of 100 m/s in a direction 30 degrees NE relative to the air. • The wind is blowing 40 m/s in a direction 45 degrees SE relative to the ground. • Find the resultant velocity of the airplane relative to the ground. 7.2 Projectile Motion Vx • When we drop a ball from a height we know that its speed increases as it falls. • The increase in speed is due to the acceleration gravity, g = 9.8 m/sec2. Vy y x 7.2 Horizontal Speed • The ball’s horizontal velocity remains constant while it falls because gravity does not exert any horizontal force. • Since there is no force, the horizontal acceleration is zero (ax = 0). • The ball will keep moving to the right at 5 m/sec. 7.2 Horizontal Speed • The horizontal distance a projectile moves can be calculated according to the formula: 7.2 Vertical Speed • The vertical speed (vy) of the ball will increase by 9.8 m/sec after each second. • After one second has passed, vy of the ball will be 9.8 m/sec. • After the 2nd second has passed, vy will be 19.6 m/sec and so on. 7.2 Calculate using projectile motion • A stunt driver steers a car off a cliff at a speed of 20 meters per second. • He lands in the lake below two seconds later. • Find the height of the cliff and the horizontal distance the car travels. 7.2 Projectiles Launched at an Angle • A soccer ball kicked off the ground is also a projectile, but it starts with an initial velocity that has both vertical and horizontal components. *The launch angle determines how the initial velocity divides between vertical (y) and horizontal (x) directions. 7.2 Steep Angle • A ball launched at a steep angle will have a large vertical velocity component and a small horizontal velocity. 7.2 Shallow Angle • A ball launched at a low angle will have a large horizontal velocity component and a small vertical one. 7.2 Projectiles Launched at an Angle The initial velocity components of an object launched at a velocity vo and angle θ are found by breaking the velocity into x and y components. 7.2 Range of a Projectile • The range, or horizontal distance, traveled by a projectile depends on the launch speed and the launch angle. 7.2 Range of a Projectile • The range of a projectile is calculated from the horizontal velocity and the time of flight. 7.2 Range of a Projectile • A projectile travels farthest when launched at 45 degrees. 7.2 Range of a Projectile • The vertical velocity is responsible for giving the projectile its "hang" time. 7.2 "Hang Time" • • • • • You can easily calculate your own hang time. Run toward a doorway and jump as high as you can, touching the wall or door frame. Have someone watch to see exactly how high you reach. Measure this distance with a meter stick. The vertical distance formula can be rearranged to solve for time: 7.2 Projectile Motion and the Velocity Vector Key Question: Can you predict the landing spot of a projectile? *Students read Section 7.2 BEFORE Investigation 7.2 Marble’s Path Vx t=? Vy y x=? In order to solve “x” we must know “t” Y = vot – ½ g t2 vot = 0 (zero) Y = ½ g t2 2y = g t2 t2 = 2y g t = 2y g 7.3 Forces in Two Dimensions • Force is also represented in x-y components. 7.3 Force Vectors • If an object is in equilibrium, all of the forces acting on it are balanced and the net force is zero. • If the forces act in two dimensions, then all of the forces in the x-direction and y-direction balance separately. 7.3 Equilibrium and Forces • It is much more difficult for a gymnast to hold his arms out at a 45-degree angle. • To see why, consider that each arm must still support 350 newtons vertically to balance the force of gravity. 7.3 Forces in Two Dimensions • Use the y-component to find the total force in the gymnast’s left arm. 7.3 Forces in Two Dimensions • The force in the right arm must also be 495 newtons because it also has a vertical component of 350 N. 7.3 Forces in Two Dimensions • When the gymnast’s arms are at an angle, only part of the force from each arm is vertical. • The total force must be larger because the vertical component of force in each arm must still equal half his weight. 7.3 Forces and Inclined Planes • An inclined plane is a straight surface, usually with a slope. • Consider a block sliding down a ramp. • There are three forces that act on the block: – gravity (weight). – friction – the reaction force acting on the block. 7.3 Forces and Inclined Planes • When discussing forces, the word “normal” means “perpendicular to.” • The normal force acting on the block is the reaction force from the weight of the block pressing against the ramp. 7.3 Forces and Inclined Planes • The normal force on the block is equal and opposite to the component of the block’s weight perpendicular to the ramp (Fy). 7.3 Forces and Inclined Planes • The force parallel to the surface (Fx) is given by Fx = mg sinθ. 7.3 Acceleration on a Ramp • Newton’s second law can be used to calculate the acceleration once you know the components of all the forces on an incline. • According to the second law: Acceleration (m/sec2) a=F m Force (kg . m/sec2) Mass (kg) 7.3 Acceleration on a Ramp • Since the block can only accelerate along the ramp, the force that matters is the net force in the x direction, parallel to the ramp. • If we ignore friction, and substitute Newtons' 2nd Law, the net force is: Fx = m g sin θ a= F m 7.3 Acceleration on a Ramp • To account for friction, the horizontal component of acceleration is reduced by combining equations: Fx = mg sin θ - m mg cos θ 7.3 Acceleration on a Ramp • For a smooth surface, the coefficient of friction (μ) is usually in the range 0.1 - 0.3. • The resulting equation for acceleration is: 7.3 Calculate acceleration on a ramp • A skier with a mass of 50 kg is on a hill making an angle of 20 degrees. • The friction force is 30 N. • What is the skier’s acceleration? 7.3 Vectors and Direction Key Question: How do forces balance in two dimensions? *Students read Section 7.3 BEFORE Investigation 7.3 Application: Robot Navigation