Vectors: the goals Be able to define the term VECTOR and identify quantities which are vectors. Be able to add vectors by the “Head to Tail Method” Be able to use the idea of vectors to solve problems in physics.
Vectors: the goals Understand that VECTORS are to be used in describing situations that involve FORCES and VELOCITIES. Understand that VECTORS will be used again when we discuss Gravitational, Electric, and Magnetic Force fields
What is a vector? A B
A B Which Ball is moving the fastest?
What is a vector? A B
A B Now try again… Which ball is moving faster?
What is a vector? A B Now try again… Which ball is moving faster? Could we say that “A” is moving twice as fast as “B”? How could we justify that?
Vectors A vector is any quantity that has a direction as well as an amount.
Definition of a vector: A vector is anything that has direction and value. Examples are: force, velocity, acceleration, momentum. Things which are NOT vectors such as mass, area, volume, are called scalars.
Using Vectors to solve a problem:
V train = 10 m/s V man = 3 m/s
Using Vectors to solve a problem: V train = 10 m/s V man = 3 m/s How fast is the man moving with respect to the ground? Which direction?
Adding vectors to get an answer Draw the first vector. Be sure to leave the length and direction exactly as it was.
Adding vectors to get an answer Draw the second vector. Be sure to leave the length and direction exactly as it was, and draw the second vector with its tail on the head of the first.
Adding vectors to get an answer The answer is called the RESULTANT. The answer is drawn from the tail of the first vector to the head of the last vector.
Adding vectors to get an answer Measure the length of the RESULTANT vector. In this case, the answer will be 7 m/s to the right.
Using Vectors to solve a problem: V train = 10 m/s V man = 3 m/s Now, how fast is the man moving with respect to the ground? In which direction?
Adding vectors to get an answer Draw the first vector. Be sure to leave the length and direction exactly as it was.
Adding vectors to get an answer Draw the second vector. Be sure to leave the length and direction exactly as it was, and draw the second vector with its tail on the head of the first.
Adding vectors to get an answer Measure the length of the RESULTANT vector. In this case, the answer will be 13 m/s to the right.
Using Vectors to solve a problem: What if the railroad car is moving at 10 m/s as before. The man on the car is walking at 3 m/s toward the side of the car. What will be the velocity of the man with respect to the ground?
Using Vectors to solve a problem:
Draw the vectors Head to tail:
Using Vectors to solve a problem: Draw the vectors Head to tail: And then draw the RESULTANT from The Tail of the First to the Head of the Last.
Using Vectors to solve a problem: Draw the vectors Head to tail: And then draw the RESULTANT from The Tail of the First to the Head of the Last. Try this yourself and tell me what you get as an answer.
Using Vectors to solve a problem: Draw the vectors Head to tail: And then draw the RESULTANT from The Tail of the First to the Head of the Last. The answer is m/s at 17.5 o from the first vector.
Using Vectors to solve a problem: What if the railroad car is moving at 10 m/s as before to the East. There is a second car on top of the first moving South at 5 m/s The man is on the top car walking at 3 m/s to the Southwest What will be the velocity of the man with respect to the ground?
Using Vectors to solve a problem:
The answer is: 10.5 m/s at an angle of 42.5 degrees south of East.
The Boat Problem A B
A B V boat = 12 m/s
The Boat Problem A B V boat = 12 m/s V river = 5 m/s
The Boat Problem A B V boat = 12 m/s V river = 5 m/s Distance = 360 meters
The Boat Problem A B What will the actual velocity of the boat be (find both speed and direction)? How long will it take to cross the river? Where will the boat land (how far downstream from point “B” will the boat land?)
The Boat Problem A B V boat = 12 m/s V river = 5 m/s
The Boat Problem A B V boat = 12 m/s V river = 5 m/s Draw the vectors head to tail.
The Boat Problem A B V boat = 12 m/s V river = 5 m/s
The Boat Problem A B V boat = 12 m/s V river = 5 m/s V actual =13 o
The Boat Problem A B V boat = 12 m/s V river = 5 m/s Distance = 360 meters
The Boat Problem A B V boat = 12 m/s V river = 5 m/s Distance = 360 meters What angle must the pilot use in order to cross the river straight, and land at point B?
The Boat Problem A B V boat = 12 m/s V river = 5 m/s The solution to this problem has to be constructed differently.
The Boat Problem V boat = 12 m/s V river = 5 m/s Draw the River velocity vector first.
The Boat Problem V boat = 12 m/s V river = 5 m/s Draw a line in the direction of the answer you want. The Head of the RESULTANT will be on the head of the V river.
The Boat Problem V boat = 12 m/s V river = 5 m/s The Head of the RESULTANT will be on the head of the V river. Rotate and fit in the 12 cm long vector into the only place it will fit. This will define where the V actual vector will be.
The Boat Problem V boat = 12 m/s V river = 5 m/s The Head of the RESULTANT will be on the head of the V river. Rotate and fit in the 12 cm long vector into the only place it will fit. This will define where the V actual vector will be. V actual
The Boat Problem A B What will the actual velocity of the boat be? What angle must be used to head the boat? How long will it take to cross the river?
The Boat Problem V boat = 12 m/s V river = 5 m/s V actual = 10.9 m/s Angle=24.6 o
The Boat Problem A B V boat = 12 m/s V river = 5 m/s Distance = 360 meters V actual =10.9 m/s
The Boat Problem A B V boat = 12 m/s V river = 5 m/s Distance = 360 meters V actual =10.9 m/s The time to get across will be 360m/10.9 m/s =33 seconds
The Boat Problem A B V boat = 12 m/s V river = 5 m/s Distance = 360 meters V actual =10.9 m/s …and the boat winds up straight across the river at point B