Motion in Two DimensionsChapter 6

Motion in Two Dimensions

In this chapter we’ll use what we learned about resolving vectors into their x- and y-components to study three types of two-dimensional motion.

Projectile motion

A form of motion in which an object or particle (called a projectile) is thrown or launched near the Earth's surface, and it moves along a curved path under the action of gravity only.

Uniform circular motion

The movement of an object at a constant speed around a circle with a fixed radius.

Relative motion

A situation in which an object is moving in a coordinate system that itself is moving,

Components of Vectors We can also resolve a vector

into two component vectors that are perpendicular to one another.

1. Draw vertical and horizontal lines from the tail of the vector.

2. Draw the rectangle that encloses the vector.

3. The sides of the rectangle are the component vectors.

So is the component of in the y-direction and is the component of in the x-direction.

The magnitudes are

𝑌

�⃗�

𝑉

𝜃

Projectile Motion

We can resolve the motion of a projectile (say a cannonball, a shot put, a satellite, etc.) into its horizontal and vertical components. If we can neglect friction and air resistance, the horizontal

component is just motion with constant velocity, since no horizontal force is acting on the object.

Similarly, the vertical component is just that of an object in free fall, i.e., a constant acceleration due to the gravitational force (which is vertical).

Example of Projectile Motion: Shot Put

Shot Put Video

Components of Projectile Motion

Projectile Motion Parameters

Example Problem - 1

52. The toy car in the figure runs off the edge of a table that is 1.225 m high. The car lands 0.400 from the base of the table.a. How long did it take the car

to fall?

b. How fast was the car going on the table?

1.225 m

0.400 m

v

Example Problem - 2

53. A dart player throws a dart horizontally at 12.4 m/s. The dart hits the board 0.32 m below the height from which it was thrown. How far away is the player from the dart board?

Projectile Motion Videos

Projectile Motion in Football

Projectile Motion in Hockey

Projectile Motion in Juggling

Mythbusters

Projectile Motion Simulation

Projectile Motion Simulator

Complete handout worksheet

Uniform Circular Motion

The movement of an object at a constant speed around a circle with a fixed radius is called uniform circular motion.

r1

r2

v1

v2

r1

r2

r

v1

v2v

𝒗=∆𝒓∆ 𝑡

𝒂=∆𝒗∆ 𝑡

Velocity is always tangent to the circle.

Acceleration is constant and directed toward the center of the circle.

Centripetal Acceleration - 1

The “r” and “v” triangles are similar triangles, so the ratio of the lengths of the corresponding sides are equal.

; then divide both sides by

=

Substitute and

(directed towards center of circle)

Newton’s 2nd Law for Circular Motion: The net centripetal force on an object moving in a circle is equal to the object’s mass times the centripetal acceleration.

Centripetal Acceleration - 2

We can measure the speed of an object moving in a circle by measuring its period (), which is the time needed for the object to make one revolution.

In one revolution the object travels a distance so its speed is

Example Problem

66. A carnival clown rides a motorcycle down a ramp and around a vertical loop. If the loop has a radius of 18 m, what is the slowest speed the rider can have at the top of the loop to avoid falling? Hint: At this slowest speed the track exerts no force on the motorcycle at the top of the loop.

Relative Velocity

The relative velocity of object a to object c is the vector sum of object a’s velocity relative to object b and object b’s velocity relative to object c.

𝒗𝑎/𝑏

𝒗𝑏 /𝑐

𝒗𝑎/𝑐

Example Problem

An airplane flies due north at 150 km/hr relative to the air. There is a wind blowing at 75 km/hr to the east relative to the ground. What is the plane’s speed relative to the ground?