2D and 3D motion

1. MOTION IN 2-
DIMENSIONS AND 3-
DIMENSIONS

2. Learning Objectives:
■ Describe motion using the concept of relative velocities in 1D
and 2D. STEM_GP12KIN-Ic20
■ Extend the definition of position, velocity, and acceleration to
2D and 3D using vector representation. STEM_GP12KIN-Ic21
■ Deduce the consequences of the independence of vertical and
horizontal components of projectile motion. STEM_GP12KIN-
Ic22
■ Calculate range, time of flight, and maximum heights of
projectiles. STEM_GP12KIN-Ic23

5. The displacement is the change
of the position
Δ𝒓 = 𝒓2 - 𝒓1 = (x2 – x1) 𝒊 + (y2 – y1) 𝒋 + (z2 – z1) 𝒌.

6. We define the average velocity 𝑣av during
this interval as the displacement divided by
the time interval:

7. We now define instantaneous velocity as
the limit of the average velocity as the time
interval approaches zero, and it equals the
instantaneous rate of change of position
with time. The key difference is that
position r and instantaneous velocity v are
now both vectors:

8. The Acceleration Vectors
The acceleration of a body in space will
describe both the changes in the
magnitude and direction of its velocity. The
average acceleration is defined as:

9. The instantaneous acceleration is
also equal to the instantaneous rate of
change of velocity with time.

10. Projectile Motion
■ A projectile is any body given an initial
velocity and then follows a curved path (or
trajectory) determined by the effects of
gravitational acceleration and air resistance. A
batted baseball, a thrown football, a package
dropped from an airplane, and a bullet shot
from a rifle are all projectiles.

11. Projectile motion is always confined to a vertical plane
determined by the direction of the initial velocity. This is
because the acceleration due to gravity is purely vertical;
gravity cannot accelerate the projectile sideways. Thus
projectile motion is two-dimensional. We will call the
plane of motion the xy-coordinate plane, with the x-axis
horizontal and the y-axis vertically upward.
The components of a are,

12. Substituting ax = 0 to the equation for the horizontal
motion, we find
Vx = Vox
X = Xo + Voxt
Substituting ay = -g, the y-component of the equations
are:
Vy = Voy – gt
y – yo = ½(V + Vo)t
y – yo = Vot – ½ gt2
-2g(y-yo) = Vy2 – Voy2

13. Where
The position of the projectile at any time is
Its speed is
And direction

15. Example:
1. A Pomeranian dog is at the origin of coordinates at
time t1=0. For the time interval from t1=0 to
t2=12.0 s, the dog’s average velocity has x-
component -3.8 m/s and y-component 4.9 m/s. At
time t2=12.0 s,
a. What is the x- and y- coordinates of the dog?
b. How far is the dog from the origin?

16. Example:
2. A batter hits a baseball so that it leaves the bat with an
initial speed Vo= 37.0 m/s at an initial angle ao= 53.1
degree at a location where g=9.8 m/s2. Let’s see how we
can predict where the ball is and how it’s moving at a
certain time, how we can find the ball’s maximum height,
and how we can find the distance from home plate to
where the comes down.
a. Find the position of the ball, and the magnitude and
direction of its velocity, when t = 2.00 s. Using the
coordinate system, we want to find x, y, vx, and vy at time
t = 2.00 s.

17. Example:
3. A ball rolls horizontally off a five meter cliff at a
speed of 5 m/s
a. How long will it take for the ball to hit the ground?
b. How from the base of the cliff will the ball lang?
c. What is the final speed of the ball just before it hits
the ground?
d. What is the final velocity of the ball just before it
hits the ground?

18. CIRCULAR MOTION
Circular motion is a movement of an object
along the circumference of a circle or
rotation along a circular path.
It can be uniform, with constant angular
rate of rotation and constant speed, or non-
uniform with a changing rate of rotation.

19. UNIFORM CIRCULAR MOTION
A car moving along a circular path. If the car is in uniform circular
motion as in (c), the speed is constant and the acceleration is directed
toward the center of the circular path.

20. UNIFORM CIRCULAR MOTION

21. Relative Motion
You have no doubt observed how a car that is moving
slowly forward appears to be moving backward when
you pass it. In general, when two observers measure
the velocity of a moving body, they get different
results if one observer is moving relative to the other.
The velocity seen by a particular observer is called the
velocity relative to that observer, or simply relative
velocity.

22. Relative Velocity in One Dimension
One must take into account relative velocities to describe the
motion of an airplane in the wind or a boat in a current.
Assessing velocities involves vector addition and a useful
approach to such relative velocity problems is to think of one
reference frame as an "intermediate" reference frame in the
form:
𝒗AC = 𝒗AB + 𝒗BC
Put into words, the velocity of A with respect to C is equal to
the velocity of A with respect to B plus the velocity of B with
respect to C. Reference frame B is the intermediate reference
frame. This approach can be used with
the airplane or boat examples.

23. Example:
1. Passengers on a carnival ride move at a
constant speed in a horizontal circle of
radius 5.0 m, making a complete circle
in 4.0 s. What is their speed and
acceleration?

24. Example:
2. Justin is driving his 1500-kg Camaro
through a horizontal curve on a level
roadway at a speed of 23 m/s. the turning
radius of the curve is 65 m. Determine the
minimum value of the coefficient of friction
which would be required to keep Justin’s
car on the curve.

25. Activity:
1. A motorcycle stunt rider rides off the edge of the
cliff. Just at the edge, his velocity is horizontal, with
the magnitude of 9.0 m/s. After 0.5 seconds, what
is the motorcycle’s
a. x and y position
b. Distance from the edge of the cliff?
c. Velocity after 0.5 s?

26. Activity:
2. A Ferris wheel of Radius 12m is turning about a
horizontal axis through its center, such that the linear
speed of a passenger on the rim is constant and
equal to 9m/s.
a. What are the magnitude and direction of the
acceleration of the passenger as he passes
through the lowest point in a circular motion?
b. How long does it take for the Ferris wheel to make
one revolution?