Chapter 3Motion in two or more dimensions

Two dimensional motion

Vector Notation

Example

Example

The vertical and horizontal components decouple

Vertical and horizontal decouple

1D equation splits into 2D

Similarly for the velocity:

Free Fall (Projectile Motion)

Quiz 1-2 Feb 21

The General Picture

A batted baseballA baseball flies at speed v0=37m/s at an angle of α0=53.1°. Find:

a)The position, velocity (magnitude and direction) of the ball at t=2s.

b)The time when the ball reaches the highest point and the height at that point.

c)The horizontal range (the horizontal distance form the starting point to where the ball hits the ground).

First find the initial velocity v0x, v0y.

v0x

v0y

Velocity (magnitude and direction)

The highest point

Horizontal rangeTo find the horizontal range, we have to first figure out when the ball hits the ground.

Different initial and final heights

You toss a ball from your window 8m above the ground. The ball leaves your hand at 10m/s at an angle of 20° below the horizontal. How far horizontally will it hit the ground?

First find the initial velocity v0x, v0y.

v0x

v0y

Find the time it hits the ground

Find the horizontal distance

Circular motion

Acceleration of Circular Motion

Centripetal acceleration

r

v

x

y

(x,y)

• Even though the speed of the object is constant, the velocity is constantly changing because of the changing direction. That is why the acceleration is non-zero.

Must remember this:

Using this you can convert freely among ω, T, f and v.

Centripetal force and acceleration

Uniform Circular MotionA fighter pilot flying in a circular turn will pass out if the centripetal acceleration he experiences is more than about 9 times the acceleration of gravity g. If his F18 is moving with a speed of 300 m/s, what is the approximate diameter of the tightest turn this pilot can make and survive to tell about it ?

(a) 500 m

(b) 1000 m

(c) 2000 m

Solution

2km