Physics 2211: Lectures 12

1
Physics 2211 Lectures 12

• 2-D, 3-D Kinematics and Projectile Motion
• Independence of x , y and z- components
• Georgia Tech track and field example
• Football example
• Shoot the monkey
• Basketball Free Throws

2
Kinematics in Two Dimensions - Review
3

• acceleration of the car is zero
• motion of the car is one-dimensional
• velocity of the car does not change
• magnitude of the acceleration is constant
• acceleration is increasing with time
• none of the above

A car moves along a circle with constant speed.
Which statement is correct?
4
Roller Coaster
5
Position and Velocity
displacement
6
Position and Velocity
average velocity vector
7
Instantaneous velocity
8
Components of velocity
9
Instantaneous Kinematic Quantities
10
3-D Kinematics

• The position, velocity, and acceleration of a
  particle in 3 dimensions can be expressed as

• We have already seen the 1-D kinematics equations

11
3-D Kinematics

• For 3-D, we simply apply the 1-D equations to
  each of the component equations.

• Which can be combined into the vector equations

12
3-D Kinematics

• So for constant acceleration we can integrate to
  get

• Aside the 4th kinematics equation can be
  written as

(more on this later)
13
2-D Kinematics

• Most 3-D problems can be reduced to 2-D problems
  when acceleration is constant
• Choose y axis to be along direction of
  acceleration
• Choose x axis to be along the other direction
  of motion
• Example Throwing a baseball (neglecting air
  resistance)
• Acceleration is constant (gravity)
• Choose y axis up ay -g
• Choose x axis along the ground in the direction
  of the throw

14
Example
15
Example

• Treat horizontal motion and vertical motion
  separately

• Then just add the results Principle of
  Superposition

• Velocity at highest point

16
Example

• Vertical Motion

• Need to determine magnitude of take-off velocity

17
Example

• Magnitude of take-off velocity

18
Example

• Two footballs are thrown from the same point on a
  flat field. Both are thrown at an angle of 30o
  above the horizontal. Ball 2 has twice the
  initial speed of ball 1. If ball 1 is caught a
  distance D1 from the thrower, how far away from
  the thrower D2 will the receiver of ball 2 be
  when he catches it?

(1) D2 2D1 (2) D2 4D1 (3) D2 8D1
19
Example

• The horizontal distance a ball will go is simply
  x (horizontal speed) x (time in air) v0x t

• To figure out time in air, consider the
  equation for the height of the ball

• When the ball is caught, y y0

(time of catch)
(time of throw)
20
Example

• So the time spent in the air is

• The range, R, is thus

• Ball 2 will go 4 times as far as ball 1!

• Notice For maximum range,

21
Shooting the Monkey(tranquilizer gun)

• Where does the zookeeper aim if he wants to hit
  the monkey?
• ( He knows the monkey willlet go as soon as he
  shoots ! )

22
Shooting the Monkey

• If there were no gravity, simply aim

at the monkey
23
Shooting the Monkey

• With gravity, still aim at the monkey!

24
Shooting the Monkey
x v0 t y -1/2 g t2

• This may be easier to think about.
• Its exactly the same idea!!

x x0 y -1/2 g t2
25
ExampleProjectile Motion Basketball Free
Throws(or life at the charity stripe)

• Projectile motion for fun and profit!

26
ExampleProjectile Motion Basketball Free
Throws

• If the basketball is thrown at 55 degrees above
  the horizontal, what must its initial speed be
  for the foul shot to go in?

27
Example
Constant Acceleration Problem

• Choose y axis up.
• Choose x axis parallel to the floor in the
  direction of the basket.
• Choose the origin (0,0) to be at the ball.

28
Example

• Find the x and y components of initial velocity

29
Example
30
Problem

• Barry Bonds clobbers a fastball toward
  center-field. The ball is hit 1 m (d ) above the
  plate, and its initial velocity is 36.5 m/s (v0 )
  at an angle of 30o (? ) above horizontal. The
  center-field wall is 113 m (D) from the plate and
  is 3 m (h) high.
• What time does the ball reach the fence?
• Does Barry get a home run?

31
Problem

• Choose y axis up.
• Choose x axis along the ground in the direction
  of the hit.
• Choose the origin (0,0) to be at the plate.
• The ball is hit at t 0, x x0 0 and y y0
  d.

Kinematics equations are
32
Problem

• Use geometry to figure out v0x and v0y

33
Problem

• The time to reach the wall is t D / v0x
  (easy!)
• We have an equation that tells us y(t) d v0y
  t - g t2/ 2
• So, were done....now we just plug in the
  numbers
• Find
• v0x 36.5 cos(30o) m/s 31.6 m/s
• v0y 36.5 sin(30o) m/s 18.25 m/s
• t (113 m) / (31.6 m/s) 3.58 s
• y(t) (1.0 m) (18.25 m/s)(3.58 s) - (0.5)(9.8
  m/s2)(3.58 s)2
• (1.0 65.3 - 62.8) m 3.5 m
• Since the wall is 3.0 m high, Barry gets the
  homer!!