Chapter 4 Two-Dimensional Kinematicsnsmn1.uh.edu/hpeng5/Peng04_LectureOutline.pdf · • Components of motion in the x- and y-directions can be treated independently • In projectile

Copyright © 2010 Pearson Education, Inc.

Chapter 4

Two-Dimensional Kinematics


Units of Chapter 4

• Motion in Two Dimensions

• Projectile Motion: Basic Equations

• Zero Launch Angle

• General Launch Angle

• Projectile Motion: Key Characteristics


4-1 Motion in Two Dimensions

If velocity is constant, motion is along a straight line:

Description of the motionby X and Y components

Description by magnitude and direction

Concept: vectors!


4-1 Motion in Two DimensionsMotion in the x- and y-directions should be solved separately:

Key: both position and velocity are vectors!Decompose them in x and y directions.

Independent 1D motion in both x and y directions


4-2 Projectile Motion: Basic Equations

Assumptions:

• ignore air resistance

• g = 9.81 m/s2, downward

• ignore Earth’s rotation

If y-axis points upward, acceleration in x-direction is zero and acceleration in y-direction is -9.81 m/s2



The acceleration is independent of the direction of the velocity:



These, then, are the basic equations of projectile motion:

Square and then get rid of t

( 0, )a a gx y= = −



Q: find the corresponding parameters for the equations!Write them down in black board in symbolic form.

Question 4.1a Firing Balls IA small cart is rolling at constant velocity on a flat track. It fires a ball straight up into the air as it moves. After it is fired, what happens to the ball?

a) it depends on how fast the cart is moving

b) it falls behind the cartc) it falls in front of the cartd) it falls right back into the carte) it remains at rest

Question 4.1a Firing Balls IA small cart is rolling at constant velocity on a flat track. It fires a ball straight up into the air as it moves. After it is fired, what happens to the ball?

a) it depends on how fast the cart is moving

b) it falls behind the cartc) it falls in front of the cartd) it falls right back into the carte) it remains at rest

when viewed from

train

when viewed from

ground

In the frame of reference of the cart, the ball only has a vertical component of velocity. So it goes up and comes back down. To a ground observer, both the cart and the ball have the same horizontal velocity, so the ball still returns into the cart.


4-3 Zero Launch AngleLaunch angle: direction of initial velocity with respect to horizontal


4-3 Zero Launch Angle

In this case, the initial velocity in the y-direction is zero. Here are the equations of motion, with x0 = 0 and y0 = h:


4-3 Zero Launch AngleThis is the trajectory of a projectile launched horizontally:

How to derive this?


4-3 Zero Launch AngleEliminating t and solving for y as a function of x:

This has the form y = a + bx2, which is the equation of a parabola.

The landing point can be found by setting y = 0 and solving for x:

Question 4.2 Dropping a Package

You drop a package from a plane flying at constant speed in a straight line. Without air resistance, the package will:

a) quickly lag behind the plane while falling

b) remain vertically under the plane while falling

c) move ahead of the plane while falling

d) not fall at all

You drop a package from a plane flying at constant speed in a straight line. Without air resistance, the package will::

a) quickly lag behind the plane while falling

b) remain vertically under the plane while falling

c) move ahead of the plane while falling

d) not fall at all

Both the plane and the package have

the same horizontal velocity at the

moment of release. They will maintain

this velocity in the x-direction, so they

stay aligned.

Follow-up:: what would happen if air resistance is present?

Question 4.2 Dropping a Package

Example 4-3 DROPPING A BALLA person skateboarding with a constant speed of 1.30 m/sreleases a ball from a height of 1.25 m above the ground. Given that x0=0 and y0=h=1.25m, find x and y for (a) t=0.250 s and (b) t=0.500 s. (c) Find the velocity, speed, and direction of motion of the ball at t=0.500s.

Hint: for part (c), express the velocity vector in terms of unit vectors

Example 4-3 DROPPING A BALLA person skateboarding with a constant speed of 1.30 m/s releases a ball from a height of 1.25 m above the ground. Given that x0=0 and y0=h=1.25m, find x and y for (a) t=0.250 s and (b) t=0.500 s. (c) Find the velocity, speed, and direction of motion of the ball at t=0.500s.

Example 4-3 DROPPING A BALLA person skateboarding with a constant speed of 1.30 m/s releases a ball from a height of 1.25 m above the ground. Given that x0=0 and y0=h=1.25m, find x and y for (a) t=0.250 s and (b) t=0.500 s. (c) Find the velocity, speed, and direction of motion of the ball at t=0.500s.


4-4 General Launch AngleIn general, v0x = v0 cos θ and v0y = v0 sin θ

This gives the equations of motion:


4-4 General Launch AngleSnapshots of a trajectory; red dots are at t = 1 s, t = 2 s, and t = 3 s


Hint: draw the trajectory in x-y plane,identify the equations to use and find all initial parameters.




Ask: Draw the trajectory in x-y plane?


Hint: set up the x-y coordinate systemand draw the trajectory in x-y plane.


Hint: set up the x-y coordinate systemand draw the trajectory in x-y plane.


4-5 Projectile Motion: Key Characteristics

Range: the horizontal distance a projectile travels

If the initial and final elevation are the same:

010 2

2( sin ) or sin

vv gt t

gθ θ

⎛ ⎞= = ⎜ ⎟

⎝ ⎠

Derivation:

y=0

R as x



The range is a maximum when θ = 45°:


Symmetry in projectile motion:



Hint: (1) set up the x-y coordinate system(2) draw the trajectory in x-y plane.(3) identify which equation to use.



Summary of Chapter 4

• Components of motion in the x- and y-directions can be treated independently

• In projectile motion, the acceleration is –g

• If the launch angle is zero, the initial velocity has only an x-component

• The path followed by a projectile is a parabola

• The range is the horizontal distance the projectile travels

Documents

Chapter 4 Two-Dimensional Kinematicsnsmn1.uh.edu/hpeng5/Peng04_LectureOutline.pdf · • Components of motion in the x- and y-directions can be treated independently • In projectile