2.3.1 The Position and Velocity Vectors

Question 1

The Position and Velocity Vectors

Answer 1

• For motion in two dimensions, all of the fundamental quantities of kinematics—position, displacement, velocity, and acceleration—are vectors.
• A good way to think about the motion of an object in two dimensions is in terms of its x- and y- components. The vector forms of the equations of two- or three-dimensional motion can be expressed as equations of motion in one dimension.

Question 2

note

Answer 2

• We have derived equations of kinematics that describe the motion of objects in one dimension. However, many objects move in two or three dimensions. For example, the path of a ball tossed in the air follows a two-dimensional arc. A ship navigating a river can move forward or backward and left or right. The motion of a skier moving down slope can be described in three dimensions.
• All of the fundamental quantities of kinematics that we have explored in one dimension can be written as vectors in two and three dimensions. These quantities include position,displacement, velocity, and acceleration.
• In one dimension, average velocity is .
• In two dimensions the average velocity vector is . The initial position of the lizard is , and then it moves to position. Notice that the average velocity vector () has in the same direction as , but its magnitude is different.
• The graph shows the path that the lizard follows as it moves in two dimensions. The instantaneous velocity vector is found by taking the limit as Dt approaches zero of the average velocity vector: . The direction of the instantaneous velocity vector is always tangent to the position function.
• When two vectors are equivalent, their components are equal in magnitude. You can write the equations of kinematics as vector equations and then break them into component form

Question 3

Look at the position function of an object. The arrows are the instantaneous velocity at different times. Which of the following best relates the speed of the object to the arrows at an instant t ?

Answer 3

The length of the arrow indicates the magnitude of the velocity vector, which is the speed of the object.

Question 4

How are the master equations of kinematics with constant acceleration for two-dimensional motion related to the master equations of kinematics for one-dimensional motion?

Answer 4

The two-dimensional equations are the same as the one-dimensional equations in two different directions: x and y.

Question 5

Which of the following statements is not correct?

Answer 5

All these statements are correct.

Question 6

Displacement vectors indicate how an object moves from one point to another. Which of the following statements related to the displacement vector in the diagram is not correct?

Answer 6

The displacement vector, delta r, is defined as rf - ri

Question 7

Which of the following expressions correctly relates v, delta t, rf, and ri

Answer 7

rf = ri + v delta t

Question 8

Consider two motion vectors A and B. Which of the following statements about these vectors is not correct?

Answer 8

A /= B, Ax /= Bx

Question 9

The symbol r is used to denote a position vector. Which of the following statements related to the position vector is not correct?

Answer 9

The position vector r = (2m)ti + (3m)tj + (0.1m)k describes motion in two dimensions

Question 10

Which of the following statements related to two-dimensional motion is not correct?

Answer 10

If r is multiplied by a positive number, the direction can possibly change

Question 11

Which of the following is not an example of two-dimensional motion?

Answer 11

an astronaut “walking” in space