Vectors and Direction Key Question: How do we accurately communicate length and distance?
Vectors and DirectionA scalar is a quantity thatcan be completely describedby one value: the magnitude.You can think of magnitudeas size or amount, includingunits.
Vectors and DirectionA vector is a quantity thatincludes both magnitude anddirection.Vectors require more thanone number. The information “1 kilometer, 40 degrees east of north” is an example of a vector.
Vectors and DirectionIn drawing a vector as anarrow you must choose ascale.If you walk five meterseast, your displacementcan be represented by a 5cm arrow pointing to theeast.
Vectors and DirectionSuppose you walk 5 meters east,turn, go 8 meters north, then turnand go 3 meters west.Your position is now 8 metersnorth and 2 meters east of whereyou started.The diagonal vector that connectsthe starting position with the finalposition is called the resultant.
Vectors and DirectionThe resultant is the sum of two ormore vectors added together.You could have walked a shorterdistance by going 2 m east and 8m north, and still ended up in thesame place.The resultant shows the most directline between the starting positionand 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?
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.
Finding the Magnitude of a VectorWhen you know the x- and y- components of a vector, and thevectors form a right triangle, you can find the magnitude usingthe Pythagorean theorem.
Adding VectorsWriting vectors in components make it easy to add them.
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.
Projectile Motion A projectile is an object moving in two dimensions under the influence of Earths gravity; its path is a parabola.
Projectile Motion by analyzing the horizontal and vertical motions separately
Projectile Motion The speed in the x- direction is constant; in the y-direction the object moves with constant acceleration g. This photograph shows two balls that start to fall at the same time. The one on the right has an initial speed in the x-direction. It can be seen that vertical positions of the two balls are identical at identical times, while the horizontal position of the yellow ball increases linearly.
Projectile MotionIf an object is launched at an initial angle of θ0 with thehorizontal, the analysis is similar except that the initialvelocity has a vertical component.
Trajectory The path a projectile follows is called its trajectory.
Trajectory, Range The trajectory of a thrown basketball follows a special type of arch- shaped curve called a parabola. The distance a projectile travels horizontally is called its range.
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.
Calculate magnitudeDraw the velocity vector v = (5, 5) m/sec and calculate the magnitude of the velocity (the speed), using the Pythagorean theorem.
Components of the Velocity VectorSuppose a car is driving 20meters per second.The direction of the vector is127 degrees.The polar representation ofthe velocity is v = (20m/sec, 127°).
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.
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 thewater.The water is also moving with another velocity relative tothe land.
Calculate velocity componentsAn airplane is moving at a velocity of 100 m/s in a direction 30degrees NE relative to the air.The wind is blowing 40 m/s in a direction 45 degrees SE relative tothe ground.Find the resultant velocity of the airplane relative to the ground.
Projectile Motion VxWhen we drop a ballfrom a height we know Vythat its speed increasesas it falls. yThe increase in speed isdue to the accelerationgravity, g = 9.8 m/sec2. x
Horizontal SpeedThe ball’s horizontal velocityremains constant while it fallsbecause gravity does not exertany horizontal force.Since there is no force, thehorizontal acceleration is zero(ax = 0).The ball will keep moving tothe right at 5 m/sec.
Horizontal SpeedThe horizontal distance a projectile moves canbe calculated according to the formula:
Vertical SpeedThe vertical speed (vy) of theball will increase by 9.8 m/secafter each second.After one second has passed, vyof the ball will be 9.8 m/sec.After the 2nd second haspassed, vy will be 19.6 m/secand so on.
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.
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 dividesbetween vertical (y) and horizontal (x) directions.
Steep AngleA ball launched ata steep angle willhave a largevertical velocitycomponent and asmall horizontalvelocity.
Shallow AngleA ball launched at alow angle will havea large horizontalvelocity componentand a small verticalone.
Projectiles Launched at an AngleThe 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.
Range of a ProjectileThe range, or horizontal distance, traveled by a projectiledepends on the launch speed and the launch angle.
Range of a Projectile The range of a projectile is calculated from the horizontal velocity and the time of flight.
Range of a Projectile A projectile travels farthest when launched at 45 degrees.
Range of a Projectile The vertical velocity is responsible for giving the projectile its "hang" time.
Hang TimeYou can easily calculate your own hang time.Run toward a doorway and jump as high as you can, touchingthe 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 fortime:
Projectile Motion and the Velocity Vector Key Question: Can you predict the landing spot of a projectile?