2.3.2 The Acceleration Vector

Question 1

The Acceleration Vector

Answer 1

• The vector formulation of instantaneous acceleration is
• The acceleration vector can be decomposed into a component, , parallel to the velocity vector, which describes changes in the magnitude of and a component, , perpendicular to the velocity vector, which describes changes in the direction of .
• Two-dimensional motion can be described using x- and y-component forms of the equations of kinematics, while three-dimensional motion can be described using x-, y-, and z-component forms of the same equations

Question 2

note

Answer 2

• The curve shows the path of a particle moving in two dimensions. The vector emanating from the origin is the position (or displacement) vector, and it describes the location of the particle at a given time. The other vector is the velocity vector. It describes the change in position of a particle per unit time. It is tangent to the position curve. Speed is the length of the velocity vector.
• In two and three dimensions average acceleration is expressed as the vector: . The vector that represents the change in velocity, D, begins at the tip of the initial velocity vector() and ends at the tip of the final velocity vector (), when and are placed tail-to-tail. The acceleration vector has the same direction as D, however its magnitude may be different.The instantaneous acceleration vector is calculated by taking the limit of the average acceleration vector as Dt tends to zero: .
• At any point in time a particle’s motion is described by its velocity and acceleration. The acceleration vector can be decomposed into components that are parallel () and perpendicular () to the velocity vector. If is in the opposite direction from , the particle’s speed is decreasing.If is in the same direction as , the particle’s speed is increasing. The component of acceleration perpendicular to indicates the change in the direction of .
• A useful way to think about the components of a vector is as the shadow, or projection, of a particle’s motion in the x-direction (illustration on left) and y-direction (illustration on right). You can easily think about motion on the x- or y-axis because it is one-dimensional motion and you have already derived equations of kinematics in one dimension.
• For example, you can derive the first equation of kinematics from the definition of the average acceleration vector. The vector equation can be decomposed into its x-components, , and into its y-components,. The component forms of the vector equationsimply describe one-dimensional motion.

Question 3

Which of the following is correct?

Answer 3

vfx = vix + ax delta t
vfy = viy + ay delta t

Question 4

The graph shows the position of an object in two-dimensional motion. Which of the following statements related to the graph is not correct?

Answer 4

You can graphically find the magnitude of the velocity vector using this graph.

Question 5

The acceleration vector can be assessed in terms of a || and a perpendicular. The parallel component is parallel to the direction of the velocity vector. Considering the velocity and acceleration vectors shown on the picture, which of the following statements is not correct?

Answer 5

The object is slowing down and but not changing direction.

Question 6

Suppose a train starts at point A and 35 seconds later, it reaches point B. What are the initial and final position vectors of the train?

Answer 6

ri = 50 mi + 100 mj
rf = 225mi - 125mj

Question 7

Which of the following offers the correct equation for and description of instantaneous acceleration for two-dimensional motion?

Answer 7

a = lim delta t -> 0 delta v / delta t

Instantaneous acceleration is the acceleration of an object at an instant in time

Question 8

Suppose you are examining the motion of a train. Over an interval of 25 second, you calculate that delta r = 100 mi - 250 mj. Given this information, what is the average velocity vector for this interval?

Answer 8

v = 4mi - 14mj
v = 14.6 m/s

Question 9

Suppose you are examining the motion of a train
ri = 25mi + 100mj
rf = -25mi + 125mj
What is the value for delta r?

Answer 9

delta r = -50mi + 25mj

Question 10

The drawing shows the initial and final velocity vectors for a moving object, as well as delta v. What is the direction of the average acceleration vector, a?

Answer 10

vector a points in the same direction as delta v

Question 11

Suppose you are examining the motion of a train. It starts at point A, heading east as 20,0 m/s. Eventually it reaches point B moving at 25.0 m/s due south. The entire trip between point A and point B takes 15s. Find delta v and a between points A and B

Answer 11

delta v = -20.0m/si - 25.0m/sj

a = -1.33m/s^2i - 1.67 m/s^2j