February, Week 3 Today: Chapter 1, Vectors Homework Assignment #3 - - - PowerPoint PPT Presentation

Vectors February 1, 2013 - p. 1/14

February, Week 3

Today: Chapter 1, Vectors Homework Assignment #3 - Due Today

Mastering Physics: 6 problems from chapter 2. Written Question: 2.88

Homework Assignment #4 - Due February 8

Mastering Physics: 8 problems from chapters 1 and 3. Written Question: 3.65

Box numbers can be found on webpage

Vectors February 1, 2013 - p. 2/14

Example III

y = y0 + (v0y)t + 1 2ayt2 vy = v0y + ayt v2

y = v2 0y + 2ay (x − x0)

Example: A person at the top of a building 30 m high, throws an egg upwards at 15 m/s. If air resistance is ignored:

• How fast will it be going after 3 s?
• How high, from where it was thrown, does the egg go before

coming back down?

Vectors February 1, 2013 - p. 2/14

Example III

y = y0 + (v0y)t + 1 2ayt2 vy = v0y + ayt v2

y = v2 0y + 2ay (x − x0)

Example: A person at the top of a building 30 m high, throws an egg upwards at 15 m/s. If air resistance is ignored:

• How fast will it be going after 3 s?
• How high, from where it was thrown, does the egg go before

coming back down?

• How long does it take the egg to hit the ground?

Vectors February 1, 2013 - p. 3/14

Free-Fall Exercise II

Which of the following statements about the egg hitting the ground is False?

Vectors February 1, 2013 - p. 3/14

Free-Fall Exercise II

Which of the following statements about the egg hitting the ground is False? (a) We could set y0 = 0 and y = −30 m.

Vectors February 1, 2013 - p. 3/14

Free-Fall Exercise II

Which of the following statements about the egg hitting the ground is False? (a) We could set y0 = 0 and y = −30 m. (b) We could set y0 = 30 m and y = 0.

Vectors February 1, 2013 - p. 3/14

Free-Fall Exercise II

Which of the following statements about the egg hitting the ground is False? (a) We could set y0 = 0 and y = −30 m. (b) We could set y0 = 30 m and y = 0. (c) Its velocity is zero.

Vectors February 1, 2013 - p. 3/14

Free-Fall Exercise II

Which of the following statements about the egg hitting the ground is False? (a) We could set y0 = 0 and y = −30 m. (b) We could set y0 = 30 m and y = 0. (c) Its velocity is zero. (d) We are actually considering the instant before it hits the ground, so its acceleration is still −g.

Vectors February 1, 2013 - p. 3/14

Free-Fall Exercise II

Which of the following statements about the egg hitting the ground is False? (a) We could set y0 = 0 and y = −30 m. (b) We could set y0 = 30 m and y = 0. (c) Its velocity is zero. (d) We are actually considering the instant before it hits the ground, so its acceleration is still −g. (e) Both (c) and (d) are false.

Vectors February 1, 2013 - p. 3/14

Free-Fall Exercise II

Which of the following statements about the egg hitting the ground is False? (a) We could set y0 = 0 and y = −30 m. (b) We could set y0 = 30 m and y = 0. (c) Its velocity is zero. (d) We are actually considering the instant before it hits the ground, so its acceleration is still −g. (e) Both (c) and (d) are false.

Vectors February 1, 2013 - p. 4/14

Example IV

y = y0 + (v0y)t + 1 2ayt2 vy = v0y + ayt v2

y = v2 0y + 2ay (x − x0)

Example: A man is in a hot-air balloon which takes off and rises with a constant 2.5 m/s speed. Just after take off, the man notices that he forgot his camera. A “friend" throws the camera up to him with a speed of 15 m/s. If the man is 2 m above the camera when it is thrown, how high will he be when he caches his camera?

Vectors February 1, 2013 - p. 5/14

Vectors

To describe two-dimensional (and three-dimensional) motion completely, we need to be able to indicate any arbitrary

• direction. We do this through the use of vectors.

Vectors February 1, 2013 - p. 5/14

Vectors

To describe two-dimensional (and three-dimensional) motion completely, we need to be able to indicate any arbitrary

• direction. We do this through the use of vectors.

Vector - Any physical quantity which has a magnitude and direction associated with it.

Vectors February 1, 2013 - p. 5/14

Vectors

To describe two-dimensional (and three-dimensional) motion completely, we need to be able to indicate any arbitrary

• direction. We do this through the use of vectors.

Vector - Any physical quantity which has a magnitude and direction associated with it. Magnitude - Positive number along with unit that expresses the “amount" of the vector.

Vectors February 1, 2013 - p. 5/14

Vectors

To describe two-dimensional (and three-dimensional) motion completely, we need to be able to indicate any arbitrary

• direction. We do this through the use of vectors.

Vector - Any physical quantity which has a magnitude and direction associated with it. Magnitude - Positive number along with unit that expresses the “amount" of the vector. Example: − → v =5 m/s at 37◦

Vectors February 1, 2013 - p. 5/14

Vectors

To describe two-dimensional (and three-dimensional) motion completely, we need to be able to indicate any arbitrary

• direction. We do this through the use of vectors.

Vector - Any physical quantity which has a magnitude and direction associated with it. Magnitude - Positive number along with unit that expresses the “amount" of the vector. Example: − → v =5 m/s at 37◦ Magnitude Direction given as angle

Vectors February 1, 2013 - p. 6/14

Drawing Vectors

To represent a vector, we use an arrow whose length is proportional to the magnitude.

Vectors February 1, 2013 - p. 6/14

Drawing Vectors

To represent a vector, we use an arrow whose length is proportional to the magnitude. − → A

Vectors February 1, 2013 - p. 6/14

Drawing Vectors

To represent a vector, we use an arrow whose length is proportional to the magnitude. − → A θ Standard Angle standard angle - From the positive x-axis

Vectors February 1, 2013 - p. 7/14

Vector Exercise

If − → A = 5 m/s at 37◦, which of the following drawing correctly shows − → B = 5 m/s at 135◦ and − → C = 10 m/s at 330◦? − → A

Vectors February 1, 2013 - p. 7/14

Vector Exercise

If − → A = 5 m/s at 37◦, which of the following drawing correctly shows − → B = 5 m/s at 135◦ and − → C = 10 m/s at 330◦? − → A (a) − → B − → C

Vectors February 1, 2013 - p. 7/14

Vector Exercise

If − → A = 5 m/s at 37◦, which of the following drawing correctly shows − → B = 5 m/s at 135◦ and − → C = 10 m/s at 330◦? − → A (a) − → B − → C (b) − → B − → C

Vectors February 1, 2013 - p. 7/14

Vector Exercise

If − → A = 5 m/s at 37◦, which of the following drawing correctly shows − → B = 5 m/s at 135◦ and − → C = 10 m/s at 330◦? − → A (a) − → B − → C (b) − → B − → C (c) − → B − → C

Vectors February 1, 2013 - p. 7/14

Vector Exercise

If − → A = 5 m/s at 37◦, which of the following drawing correctly shows − → B = 5 m/s at 135◦ and − → C = 10 m/s at 330◦? − → A (a) − → B − → C (b) − → B − → C (c) − → B − → C (d) − → B − → C

Vectors February 1, 2013 - p. 7/14

Vector Exercise

If − → A = 5 m/s at 37◦, which of the following drawing correctly shows − → B = 5 m/s at 135◦ and − → C = 10 m/s at 330◦? − → A (a) − → B − → C (b) − → B − → C (c) − → B − → C (d) − → B − → C (e) Both (a) and (c)

Vectors February 1, 2013 - p. 7/14

Vector Exercise

If − → A = 5 m/s at 37◦, which of the following drawing correctly shows − → B = 5 m/s at 135◦ and − → C = 10 m/s at 330◦? − → A (a) − → B − → C (b) − → B − → C (c) − → B − → C (d) − → B − → C (e) Both (a) and (c)

Vectors February 1, 2013 - p. 8/14

Vector Exercise Followup

If − → A = 5 m/s at 37◦, which of the following drawing correctly shows − → B = 5 m/s at 135◦ and − → C = 10 m/s at 330◦? − → A − → B − → C − → B − → C

Vectors February 1, 2013 - p. 8/14

Vector Exercise Followup

If − → A = 5 m/s at 37◦, which of the following drawing correctly shows − → B = 5 m/s at 135◦ and − → C = 10 m/s at 330◦? − → A − → B − → C Equal length to − → A 2× longer than − → A − → B − → C

Vectors February 1, 2013 - p. 8/14

Vector Exercise Followup

If − → A = 5 m/s at 37◦, which of the following drawing correctly shows − → B = 5 m/s at 135◦ and − → C = 10 m/s at 330◦? − → A − → B − → C Equal length to − → A 2× longer than − → A − → B − → C Equal length to − → A 2× longer than − → A

Vectors February 1, 2013 - p. 8/14

Vector Exercise Followup

If − → A = 5 m/s at 37◦, which of the following drawing correctly shows − → B = 5 m/s at 135◦ and − → C = 10 m/s at 330◦? − → A − → B 135◦ − → C 330◦ Equal length to − → A 2× longer than − → A − → B − → C Equal length to − → A 2× longer than − → A

Vectors February 1, 2013 - p. 8/14

Vector Exercise Followup

If − → A = 5 m/s at 37◦, which of the following drawing correctly shows − → B = 5 m/s at 135◦ and − → C = 10 m/s at 330◦? − → A − → B 135◦ − → C 330◦ Equal length to − → A 2× longer than − → A − → B − → C 135◦ Equal length to − → A 2× longer than − → A

Vectors February 1, 2013 - p. 8/14

Vector Exercise Followup

If − → A = 5 m/s at 37◦, which of the following drawing correctly shows − → B = 5 m/s at 135◦ and − → C = 10 m/s at 330◦? − → A − → B 135◦ − → C 330◦ Equal length to − → A 2× longer than − → A − → B − → C 330◦ 135◦ Equal length to − → A 2× longer than − → A

Vectors February 1, 2013 - p. 8/14

Vector Exercise Followup

If − → A = 5 m/s at 37◦, which of the following drawing correctly shows − → B = 5 m/s at 135◦ and − → C = 10 m/s at 330◦? − → A − → B 135◦ − → C 330◦ Equal length to − → A 2× longer than − → A − → B − → C 330◦ 135◦ Equal length to − → A 2× longer than − → A Two vectors are equal when they have the same magnitude and angle, regardless of where they start

Vectors February 1, 2013 - p. 9/14

Scalar Multiplication

Multiplying a vector by a scalar changes the magnitude but not the direction of a vector.

Vectors February 1, 2013 - p. 9/14

Scalar Multiplication

Multiplying a vector by a scalar changes the magnitude but not the direction of a vector. Example: − → A = 5 m/s at 37◦, 3− → A =?

Vectors February 1, 2013 - p. 9/14

Scalar Multiplication

Multiplying a vector by a scalar changes the magnitude but not the direction of a vector. Example: − → A = 5 m/s at 37◦, 3− → A = 15 m/s at 37◦

Vectors February 1, 2013 - p. 9/14

Scalar Multiplication

Multiplying a vector by a scalar changes the magnitude but not the direction of a vector. Example: − → A = 5 m/s at 37◦, 3− → A = 15 m/s at 37◦ One “exception": Negative numbers change magnitude and flip direction by 180◦.

Vectors February 1, 2013 - p. 9/14

Scalar Multiplication

Multiplying a vector by a scalar changes the magnitude but not the direction of a vector. Example: − → A = 5 m/s at 37◦, 3− → A = 15 m/s at 37◦ One “exception": Negative numbers change magnitude and flip direction by 180◦. − → A −3− → A =?

Vectors February 1, 2013 - p. 9/14

Scalar Multiplication

Multiplying a vector by a scalar changes the magnitude but not the direction of a vector. Example: − → A = 5 m/s at 37◦, 3− → A = 15 m/s at 37◦ One “exception": Negative numbers change magnitude and flip direction by 180◦. − → A −3− → A

Vectors February 1, 2013 - p. 9/14

Scalar Multiplication

Multiplying a vector by a scalar changes the magnitude but not the direction of a vector. Example: − → A = 5 m/s at 37◦, 3− → A = 15 m/s at 37◦ One “exception": Negative numbers change magnitude and flip direction by 180◦. − → A −3− → A Of particular interest: − → A = −− → B ⇒ equal magnitude but opposite direction

• equal but opposite

Vectors February 1, 2013 - p. 10/14

Vector Addition

Vector Addition - The net result of two or more vectors, i.e., taking direction into account while adding.

Vectors February 1, 2013 - p. 10/14

Vector Addition

Vector Addition - The net result of two or more vectors, i.e., taking direction into account while adding. There are two methods of adding vectors - the graphical and component methods.

Vectors February 1, 2013 - p. 10/14

Vector Addition

Vector Addition - The net result of two or more vectors, i.e., taking direction into account while adding. There are two methods of adding vectors - the graphical and component methods. Graphical Addition - Drawing pictures and placing the vectors, “tip-to-tail" in order to determine the vector sum.

Vectors February 1, 2013 - p. 11/14

Example II

Add the following vectors.

Vectors February 1, 2013 - p. 11/14

Example II

Add the following vectors. − → A − → B

Vectors February 1, 2013 - p. 11/14

Example II

Add the following vectors. − → A − → B Vectors can be drawn at any point. As long as the magnitude and direction don’t change.

Vectors February 1, 2013 - p. 11/14

Example II

Add the following vectors. First draw − → A.

Vectors February 1, 2013 - p. 11/14

Example II

Add the following vectors. Then draw − → B at the front of − → A.

Vectors February 1, 2013 - p. 11/14

Example II

Add the following vectors. − → R = − → A + − → B The vector sum

• r resultant, −

→ R goes from the remaining tail to tip.

Vectors February 1, 2013 - p. 11/14

Example II

Add the following vectors. − → R = − → A + − → B The vector sum

• r resultant, −

→ R goes from the remaining tail to tip. A carefully drawn picture can give magnitude and direction of − →

• R. Simply use a ruler and protractor.

Vectors February 1, 2013 - p. 12/14

Vector Addition is commutative

You can add vectors in either order and the answer is the same! − → R = − → A + − → B = − → B + − → A

Vectors February 1, 2013 - p. 12/14

Vector Addition is commutative

You can add vectors in either order and the answer is the same! − → R = − → A + − → B = − → B + − → A − → A − → B

Vectors February 1, 2013 - p. 12/14

Vector Addition is commutative

You can add vectors in either order and the answer is the same! − → R = − → A + − → B = − → B + − → A First do − → A + − → B.

Vectors February 1, 2013 - p. 12/14

Vector Addition is commutative

You can add vectors in either order and the answer is the same! − → R = − → A + − → B = − → B + − → A First do − → A + − → B.

Vectors February 1, 2013 - p. 12/14

Vector Addition is commutative

You can add vectors in either order and the answer is the same! − → R = − → A + − → B = − → B + − → A First do − → A + − → B.

Vectors February 1, 2013 - p. 12/14

Vector Addition is commutative

You can add vectors in either order and the answer is the same! − → R = − → A + − → B = − → B + − → A Now do − → B + − → A.

Vectors February 1, 2013 - p. 12/14

Vector Addition is commutative

You can add vectors in either order and the answer is the same! − → R = − → A + − → B = − → B + − → A Now do − → B + − → A.

Vectors February 1, 2013 - p. 13/14

Vector Addition Exercise

For the vectors − → A and − → B, which of the following correctly shows − → R, where − → R = − → A + − → B? − → A − → B

Vectors February 1, 2013 - p. 13/14

Vector Addition Exercise

For the vectors − → A and − → B, which of the following correctly shows − → R, where − → R = − → A + − → B? − → A − → B (a) − → R

Vectors February 1, 2013 - p. 13/14

Vector Addition Exercise

For the vectors − → A and − → B, which of the following correctly shows − → R, where − → R = − → A + − → B? − → A − → B (a) − → R (b) − → R

Vectors February 1, 2013 - p. 13/14

Vector Addition Exercise

For the vectors − → A and − → B, which of the following correctly shows − → R, where − → R = − → A + − → B? − → A − → B (a) − → R (b) − → R (c) − → R

Vectors February 1, 2013 - p. 13/14

Vector Addition Exercise

For the vectors − → A and − → B, which of the following correctly shows − → R, where − → R = − → A + − → B? − → A − → B (a) − → R (b) − → R (c) − → R (d) − → R

Vectors February 1, 2013 - p. 13/14

Vector Addition Exercise

For the vectors − → A and − → B, which of the following correctly shows − → R, where − → R = − → A + − → B? − → A − → B (a) − → R (b) − → R (c) − → R (d) − → R (e) − → R

Vectors February 1, 2013 - p. 13/14

Vector Addition Exercise

For the vectors − → A and − → B, which of the following correctly shows − → R, where − → R = − → A + − → B? − → A − → B (a) − → R (b) − → R (c) − → R (d) − → R (e) − → R

Vectors February 1, 2013 - p. 13/14

Vector Addition Exercise

For the vectors − → A and − → B, which of the following correctly shows − → R, where − → R = − → A + − → B? − → A − → B − → R (a) − → R (b) − → R (c) − → R (d) − → R (e) − → R

Vectors February 1, 2013 - p. 13/14

Vector Addition Exercise

For the vectors − → A and − → B, which of the following correctly shows − → R, where − → R = − → A + − → B? − → A − → B − → R (a) − → R (b) − → R (c) − → R (d) − → R (e) − → R

Vectors February 1, 2013 - p. 14/14

Vector Subtraction

The previous example contained two vector subtractions. − → A − → B

Vectors February 1, 2013 - p. 14/14

Vector Subtraction

The previous example contained two vector subtractions. − → A − → B These are both − → B − − → A

Vectors February 1, 2013 - p. 14/14

Vector Subtraction

The previous example contained two vector subtractions. − → A − → B These are both − → B − − → A − → A − → B Traditionally: − → B − − → A = − → B +

• −−

→ A

Vectors February 1, 2013 - p. 14/14

Vector Subtraction

The previous example contained two vector subtractions. − → A − → B These are both − → B − − → A − → A − → B Traditionally: − → B − − → A = − → B +

• −−

→ A

Vectors February 1, 2013 - p. 14/14

Vector Subtraction

The previous example contained two vector subtractions. − → A − → B These are both − → B − − → A − → A − → B Traditionally: − → B − − → A = − → B +

• −−

→ A

Vectors February 1, 2013 - p. 14/14

Vector Subtraction

The previous example contained two vector subtractions. − → A − → B These are both − → B − − → A − → A − → B Traditionally: − → B − − → A = − → B +

• −−

→ A

Vectors February 1, 2013 - p. 14/14

Vector Subtraction

The previous example contained two vector subtractions. − → A − → B These are both − → B − − → A − → A − → B Traditionally: − → B − − → A = − → B +

• −−

→ A

Vectors February 1, 2013 - p. 14/14

Vector Subtraction

The previous example contained two vector subtractions. − → A − → B These are both − → B − − → A − → A − → B Traditionally: − → B − − → A = − → B +

• −−

→ A

• −

→ A − → B Less Traditional: − → B − − → A From − → A to − → B

Vectors February 1, 2013 - p. 14/14

Vector Subtraction

The previous example contained two vector subtractions. − → A − → B These are both − → B − − → A − → A − → B Traditionally: − → B − − → A = − → B +

• −−

→ A

• −

→ A − → B Less Traditional: − → B − − → A From − → A to − → B

Vectors February 1, 2013 - p. 14/14

Vector Subtraction

The previous example contained two vector subtractions. − → A − → B These are both − → B − − → A − → A − → B Traditionally: − → B − − → A = − → B +

• −−

→ A

• −

→ A − → B Less Traditional: − → B − − → A From − → A to − → B

Vectors February 1, 2013 - p. 14/14

Vector Subtraction

The previous example contained two vector subtractions. − → A − → B These are both − → B − − → A − → A − → B Traditionally: − → B − − → A = − → B +

• −−

→ A

• −

→ A − → B Less Traditional: − → B − − → A From − → A to − → B All purple vectors are equal!