Lecture 4: Motion in two dimensions
• Position, velocity, and acceleration in 2-d
• Separation of motion in x-and y-direction
• Equations for 2-d kinematics at constant acceleration
• Projectile motion
Velocity
Position vector 𝑟 = 𝑟𝑥 𝑖 + 𝑟𝑦 𝑗 = 𝑥𝑖 + 𝑦𝑗
Average velocity 𝑣𝑎𝑣 =
∆𝑟
∆𝑡
Instantaneous velocity: 𝑣 =
𝑣 = 𝑣𝑥 𝑖 + 𝑣𝑦 𝑗 =
𝑑𝑥
𝑖
𝑑𝑡
+
𝑑𝑦
𝑗
𝑑𝑡
𝑑𝑟
𝑑𝑡
𝑣𝑥 =
𝑑𝑥
𝑑𝑡
, 𝑣𝑦 =
𝑑𝑦
𝑑𝑡
Small change of
position vector in the
direction of velocity
vector
𝑟𝑓 = 𝑟𝑖 + 𝑣𝑑𝑡
Acceleration
Particle has velocity vector 𝑣 = 𝑣𝑥 𝑖 + 𝑣𝑦 𝑗 =
𝑑𝑣
Acceleration: 𝑎 =
𝑑𝑡
𝑑𝑣𝑥
𝑎 = 𝑎𝑥 𝑖 + 𝑎𝑦 𝑗 =
𝑖
𝑑𝑡
𝑎𝑥 =
𝑑𝑣𝑥
𝑑𝑡
=
+
𝑑𝑣𝑦
𝑑𝑡
𝑑2 𝑥
𝑑𝑡 2
𝑑𝑥
𝑖
𝑑𝑡
+
𝑑𝑦
𝑗
𝑑𝑡
𝑗
, 𝑎𝑦 =
𝑑𝑣𝑦
𝑑𝑡
=
𝑑2 𝑦
𝑑𝑡 2
Small change in velocity vector
occurs in the direction of the
acceleration vector
𝑣𝑓 = 𝑣𝑖 + 𝑎𝑑𝑡
Acceleration changes velocity, i.e.
speed and direction of motion.
Effect of acceleration components
Components of acceleration parallel and perpendicular to
velocity have different effects.
𝑑 𝑣 = 𝑎𝑑𝑡
𝑎II causes change in magnitude of velocity vector (speed)
𝑎┴ causes change in direction
Demonstrations
• Vertical launch of ball from traveling car
• Simultaneously dropped and horizontally
launched balls
Kinematics equations
For constant acceleration:
𝑥 = 𝑥0 + 𝑣0𝑥 𝑡 + ½𝑎𝑥 𝑡 2
𝑦 = 𝑦0 + 𝑣0𝑦 𝑡 + ½𝑎𝑦 𝑡 2
𝑣𝑥 = 𝑣0𝑥 + 𝑎𝑥 𝑡
𝑣𝑦 = 𝑣0𝑦 + 𝑎𝑦 𝑡
2
𝑣𝑥2 = 𝑣0𝑥
+ 2𝑎𝑥 (𝑥 − 𝑥0 )
2
𝑣𝑦2 = 𝑣0𝑦
+ 2𝑎𝑦 (𝑦 − 𝑦0 )
Projectile Motion
If only gravity acts on an object (free fall), then
acceleration is a constant vector of magnitude g,
directed down.
Effect on velocity:
𝑣𝑥 = 𝑣0𝑥 + 𝑎𝑥 𝑡 = 𝑣0𝑥
𝑣𝑦 = 𝑣0𝑦 + 𝑎𝑦 𝑡 = 𝑣0𝑦 − 𝑔𝑡
Projectile motion: Simulation
http://www.walter-fendt.de/ph14e/projectile.htm
Free-fall trajectory
𝑥 = 𝑥0 + 𝑣0𝑥 𝑡 + ½𝑎𝑥 𝑡 2
Worked out on the board…
𝑦 = 𝑦0 + 𝑣0𝑦 𝑡 + ½𝑎𝑦 𝑡 2
Example
A man is stranded between a
river and a high vertical cliff. To
get help, he wants to throw a
bottle containing a message over
the river. If he throws the bottle
with an initial velocity V0 and at a
positive angle θ with respect to
the horizontal, what is the
minimum height H he has to
climb up the cliff to ensure that
the bottle just barely reaches the
opposite river bank, a distance D
away?
Demo: The hunter and the monkey
y
H
V0
D
x
*You will work this out in the Homework.
Hint: the angle θ between initial velocity and horizontal is not given,
but knowing D and H will enable you to find sin θ and cos θ.