Chapter 3 - Motion in Two and Three Dimensions

I.In order to describe the motion of an object, you need to be able to describe:

A.Where is the object located? Use the position vector , where , and x, y, and z are the coordinates. (Where is the object if , in meters?)

Suppose the object is at position P1 at time t1 and position P`2 at time t2, where at P1 and at P2.

B.How far did the object move?

1.displacement,

2.distance, s = path length

C.How fast is the object moving? - velocity?

1.average velocity,

What is the direction of the average velocity?

2.instantaneous velocity, ,

so

What is the direction of the instantaneous velocity, ?

D.How fast is the velocity changing? – acceleration?

1.average acceleration,

What is the direction of the average acceleration?

2.instantaneous acceleration, , where and

What is the direction of the instantaneous acceleration?

3.Accelerations parallel and perpendicular to the motion

Suppose an object is traveling with varying velocities along a curved path. The velocity vector is tangent to the path, and the acceleration vector tells you the direction in which the velocity is changing.

Velocity vector and acceleration vector parallel:

Velocity vector and acceleration vector antiparallel:

Velocity vector and acceleration vector perpendicular:

Velocity vector and acceleration vector making an acute angle with each other:

Velocity vector and acceleration vector making an obtuse angle with each other:

E.Example: Suppose the position of an object moving in the x-y plane is described by the following expression: in the SI system of units. Find positions, velocities, and accelerations at the times t = 0 and t = 3 sec.

II.Projectile Motion - motion of an object close to the surface of the earth where air resistance is negligible.

A.Examine the motion of a projectile traveling off the end of a horizontal surface

B.observations:

1.the object travels in a curved path

2.the horizontal component of the motion indicates a constant horizontal velocity

3.the vertical component of the motion is identical to that of a freely falling body

C.Conclusions: ax = 0 and ay = -g

D.Equations for projectile motion: Both the x and y components of the acceleration are constant. So we can use the equations of constant acceleration that we have already derived in Chapter 2.

Equations of Constant Acceleration / Projectile Motion Equations where
x = x0 and y = y0 when t = 0
Horizontal motion / Vertical motion

Note that x0 = 0 and y0 = 0 when the origin is at the initial position of the projectile.

E.Path of the projectile

v0 = initial velocity of the projectile

0 = initial angle of projection of the projectile

F.Examples:

1.Mathematically, show that the path of a projectile is parabolic.

2.A ball is thrown horizontally with an initial velocity of 20 m/s from the top of a 30 m high building.

a.How much time will it take the ball to hit the ground?

b.How far does the ball travel horizontally?

c.What is the velocity (magnitude and direction) of the ball as it hits the ground?

3.A ball is thrown with an initial velocity of 20 m/s at an angle of 37o above the horizontal from the top of the 30 m high building. How far does it travel horizontally?

4.Special Case: A ball is thrown with an initial velocity of v0 at an angle of 0 over level ground. What is the range of the ball? At what angle must the ball be thrown in order to get maximum range?

5.A ball is thrown with an initial velocity of 40 m/s at an angle of 37o to the horizontal. The ball hits a window in a building that is 25 m above the ground. How far away is the building?

III.Curvilinear Motion - the motion of an object along curved paths is called curvilinear motion. This motion is quite complex, but it can be analyzed as a series of circular motions. Because of this, you need a good understanding of circular motion.

A.Uniform circular motion - motion in a circular path at a constant speed. Does an object traveling in uniform circular motion have an acceleration? (Recall acceleration parallel and perpendicular to the path an object takes.)

Derive an expression for the acceleration:

Example: What is the centripetal acceleration of the moon toward the earth? The distance from the earth to the moon is 3.85 x 108 m and the time for one complete orbit is 27.3 days.

B.Nonuniform Circular Motion - an object traveling in a circular path where its speed along the circular path varies. In order to describe the speed of the object two more unit vectors must be defined: . is a unit vector directed along a radius away from the center of the circle. is a unit vector that is tangent to the circular path in which the object travels.

The acceleration is written as:

ar = = radial acceleration - exists for any motion along a curved path

at = = tangential acceleration - exists if the speed of the object along the circular path changes

1.Summarizing: If an object is traveling along a path, and

a.the direction of travel changes, then the object has a radial acceleration.

b.the speed at which it travels changes, then it has a tangential acceleration.

2.Questions:

a.Give an example of an object that has a radial acceleration, but no tangential acceleration.

b.Give an example of an object that has a tangential acceleration, but no radial acceleration.

c.Give an example of an object that has no radial and no tangential acceleration.

d.Give an example of an object that has both a radial and a tangential acceleration.

3.Example: A car starts from rest and uniformly increases in speed as it travels around a circular road with a 30 m radius. After 8 seconds the car is traveling at 10 m/s. What is the total acceleration of the car at this time?

IV.Relative Velocity

A.Frames of Reference – positions from which measurements and observations are made. They can be stationary or moving, e.g., the earth, a car, an airplane, a river, a person, etc.

B.Notation: means the position of A relative to B; or the position of A as measured by B

means the velocity of P relative to E; or the velocity of P as measured by E

Also note that means the position of B relative to A. This is a position vector that points in the opposite direction from . So, . This is also true for the velocity vectors: When the subscripts of a vector are reversed, the resulting vector is the negative of the given vector.

C.Suppose a person (P) is standing on a street corner, a VW (C) is traveling up a hill, and an airplane (A) is flying in the sky above.

Draw the position vectors of the airplane relative to the person, the car relative to the person, and the airplane relative to the car. Notice that

Note the order of the subscripts. The equation says

Now take the time derivative of the equation:

This is an equation of the relative velocities:

Notice that if you knew the airplane’s velocity relative to the ground and the car’s velocity relative to the ground, then you can find the velocity of the airplane relative to the car.

How do you find the car’s velocity relative to the airplane? Or the person’s velocity relative to the car? Or the person’s velocity relative to the airplane?

Because the relative velocity equation is a vector expression, it can be broken up into its components:

for velocities along the x-axis,

for velocities along the y-axis, and

for velocities along the z-axis. . Examples:

1.A car and truck are traveling along a straight highway. The car is traveling at 65 mi/hr to the right and the truck is traveling at 55 mi/h to the left. How fast is the truck traveling as measured by someone in the car?

2.A one-half mile wide river flows to the east at 3 mi/hr. A boat can travel at a speed of 7 mi/hr (in still water) relative to the water. If the boat is pointed directly at the opposite shore, then where on the opposite shore does it land? And how much time does it take the boat to get there?

3.In what direction must the boat be pointed if it is to travel straight across the river?

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