1. Last Time:
Vectors
Introduction to Two-Dimensional Motion
Today:
Two-Dimensional Motion:
Projectiles, Relative Velocity
HW #2 due tonight, 11:59 p.m.
HW #3 now available
Due Tuesday, Sept 21, 11:59 p.m. (~1.5 weeks)
(Last HW before Exam #1)

2. Announcements Old: New:
Minor change to syllabus (just the lecture topic for Thursday September 16). Nothing else changed.
Old:
New:

3. Announcements Schedule for Physics Resource Room posted on Blackboard.
List of potential private tutors posted on Blackboard.

4. Projectile Motion y x
Will now start studying objects that move in both the x- and y-directions simultaneously under constant acceleration (along one or more axes).
Projectile motion (motion of objects through Earth’s gravitational field) is a special case of this.
If air resistance and the Earth’s rotation are neglected, the path of a projectile is a parabola.

5. Projectile Motion Key Point :
vertical, y
ay = −g
horizontal, x
Key Point :
For projectiles, the horizontal and vertical motions are completely independent of each other. Motion in one direction has no impact on motion in the other direction.
Gravity influences motion only in the y-direction (vertical).
We can treat the motion in the x-direction and y-direction separately, as two separate 1D problems.

6. Projectile Motion: Key Features
initial velocity vector

7. Projectile Motion: Key Features
Trajectory is a parabola
Direction and Magnitude of velocity vector changes along the trajectory

8. Projectile Motion: Key Features
At max height, y-component of velocity = vy = 0

9. Projectile Motion Analysis
vertical, y
horizontal, x θ0
v0 : initial velocity vector
We apply the Equations of Motion we developed for 1D motion separately to the motion in the x- and y-directions.
Important Difference: The initial velocity has two components:
v0x = v0 cos θ0
v0x : initial velocity at t=0 in x-direction
v0y = v0 sin θ0
v0y : initial velocity at t=0 in y-direction

10. Projectile Motion Analysis
vertical, y
horizontal, x θ0
v0 : initial velocity vector
Motion in the x-direction :
Motion in the y-direction :

11. Projectile Motion Analysis
vertical, y
horizontal, x θ0
v0 : initial velocity vector
x-direction:
Gravity does not act in the x-direction!
So, assuming air resistance is negligible: ax = 0
vx = v0x = constant for all times, does not change during trajectory

12. Projectile Motion Analysis
vertical, y
horizontal, x θ0
v0 : initial velocity vector
y-direction:
Gravity does act in the y-direction: ay = −g

13. Projectile Motion Analysis
vertical, y
v0 : initial velocity vector v θ0
horizontal, x
Speed: (at any time)
Angle of velocity vector relative to the x-axis: (at any time)
With vx and vy calculated as functions of time according to formulas on previous slides
For a trajectory such as that sketched above, θ will be in the range of −90° to +90°.

14. Projectile Motion: Summary of Facts
Unless otherwise stated, assuming air resistance is negligible and there is no other source of acceleration along the x-axis, vx remains constant, because ax = 0. 1 2
ay = −g
The equations for vy and Δy are identical to those of a freely falling object in 1D. 3
At any time t, projectile motion can be described as a “superposition” of two independent motions in the x- and y-directions. 4
In this context, by “superposition” we mean that at any time :
The x-displacement of the object is given by Δx and the y-displacement by Δy,
which then gives the object’s (x,y) position.
The x- and y-components of the object’s velocity vector are vx and vy.

15. Example
Stephen Strasburg (Washington Nationals baseball pitcher) was clocked at miles/hour in his MLB debut. If the pitch is thrown horizontally with this velocity, how far would the ball fall vertically by the time it reaches home plate, 60.5 feet away ?
100.0 mph
60.5 feet

16. Example: “Range” of a Projectile
vertical, y
horizontal, x θ0
v0 : initial velocity vector v
Obtain an expression for the horizontal distance the projectile travels, in terms of the variables v0, θ0, and g. This distance is called the projectile’s “range”.
Challenge: What angle θ0 produces the maximum range?

17. Example 3.8 (p. 68) y
v0 = 20 m/s
30° O x
45 m
A rock is thrown off of a 45-m tall building with an initial velocity as shown. Neglect air resistance. (a) How long does it take the rock to reach the ground?
(b) Find the rock’s speed at impact.
(c) Find the rock’s horizontal range.

18. What if ax is non-zero?
So far, in working these projectile problems, we have assumed that there is no acceleration in the horizontal direction: ax = 0.
If there is a vertical acceleration and a horizontal acceleration (e.g., a hypothetical rocket engine providing acceleration only in the x-direction, while undergoing free-fall in the y-direction) these accelerations form a vector quantity with components ax and ay .
We then have to use the full equations of motion for the horizontal motion, with the value of ax .

19. Relative Velocity
Suppose driving at 77 mph, and pass a car driving at 75 mph. If you watch the other car as you pass it, your speed relative to the other car does not appear to be too fast (i.e., it seems to take a long time to pass the car !).
In general, the velocity of an object depends on the velocity of the observer relative to (or with respect to) the object.
Example:
Suppose driving at 65 mph. To observer standing on the side of the road, your velocity would appear to be 65 mph. However, to someone moving at 60 mph in the same direction as you, your velocity would only appear to be 5 mph !

20. Relative Velocity
Measurements of velocity depend on the reference frame of the observer.
Reference frames are just coordinate systems.
Most of the time, our reference frame is the Earth, but sometimes we will use a moving frame of reference associated with an object moving at constant velocity relative to the Earth.

21. Relative Velocity
In 1D, this is relatively straightforward to understand.
However, in 2D, relative velocity calculations can be confusing. Suppose we have two objects, A and B, moving relative to each other, and an observer on the Earth, E.
Assume observer E at origin. y A
rAE
Position vector of A as measured by E
rAE
rBE
Position vector of B as measured by E
rAB x E
Position of A as measured by B
rAB
rBE B

22. Relative Velocity A rAE rAB E rBE B yx y E A B
rAE
rAB
rBE
Position of A as measured by an observer on B.
Velocity of A as measured by an observer on B.
Doesn’t work for velocities with magnitudes near the speed of light (need Einstein’s theory of special relativity). See PHY 213.

23. Example 3.11 (p. 72)
This boat is heading due north as it crosses a wide river with a velocity of 10 km/hr relative to the water. The river has a uniform velocity of 5 km/hr due east. Determine the velocity of the boat with respect to an observer on the riverbank.
Moving “Observers”:
B (boat) and R (river)
Fixed “Observer”:
E (“Earth”, riverbank)

24. Reading Assignment
Next class: 4.1 – 4.4