If velocity vector of an object is 10 units along x axis at t = 0 and 10 units along y axis at t = 5 sec, write its initial
and final velocity in vector form (in terms of ij and k).
a. Subtract initial velocity from final velocity to find change in velocity in vector form.
b. Find the magnitude of change in velocity.
C. Divide the magnitude of change in velocity to find magnitude of acceleration.
Answers
Answer:
4.1 Displacement and Velocity Vectors
LEARNING OBJECTIVES
By the end of this section, you will be able to:
Calculate position vectors in a multidimensional displacement problem.
Solve for the displacement in two or three dimensions.
Calculate the velocity vector given the position vector as a function of time.
Calculate the average velocity in multiple dimensions.
Displacement and velocity in two or three dimensions are straightforward extensions of the one-dimensional definitions. However, now they are vector quantities, so calculations with them have to follow the rules of vector algebra, not scalar algebra.
Displacement Vector
To describe motion in two and three dimensions, we must first establish a coordinate system and a convention for the axes. We generally use the coordinates x, y, and z to locate a particle at point P(x, y, z) in three dimensions. If the particle is moving, the variables x, y, and z are functions of time (t):
x
=
x
(
t
)
y
=
y
(
t
)
z
=
z
(
t
)
.
The position vector from the origin of the coordinate system to point P is
→
r
(
t
)
.
In unit vector notation, introduced in Coordinate Systems and Components of a Vector,
→
r
(
t
)
is
→
r
(
t
)
=
x
(
t
)
^
i
+
y
(
t
)
^
j
+
z
(
t
)
^
k
.
(Figure) shows the coordinate system and the vector to point P, where a particle could be located at a particular time t. Note the orientation of the x, y, and z axes. This orientation is called a right-handed coordinate system (Coordinate Systems and Components of a Vector) and it is used throughout the chapter.
An x y z coordinate system is shown, with positive x out of the page, positive y to the right, and positive z up. A point P, with coordinates x of t, y of t, and z of t is shown. All of P’s coordinates are positive. The vector r of t from the origin to P is also shown as a purple arrow. The coordinates x of t, y of t and z of t are shown as dashed lines. X of t is a segment in the x y plane, parallel to the x axis, y of t is a segment in the x y plane, parallel to the y axis, and z of t is a segment parallel to the z axis.