Motion in Several Dimensions

Motion in Several Dimensions

We live in three dimension, but very often it is possible to forget about one of the direction in physics problems.

Generally, a vector in space has three components. But if motion of an object takes place in a plane, we can rotate our coordinate system so that one dimension is unimportant

Components of MotionAn object in motion on a plane can be located using only two numbers – the x and y coordinates of its position. Similarly, its velocity can be described using components along the x- and y-axes.

Ball rollin

g on pool table

Components of Motion in 2D

The velocity components are:

The magnitude of the velocity vector is:

Components of Motion in 2D

The components of the displacement are then given by:

The x and y components are calculated independently, they are completely separate.

Components of Motion in 2D

The equations of motion in 2D are:

When solving two-dimensional motion problems, each component is treated separately. The only thing the components have in common is time.

Components of Motion in 2D

If the acceleration is not parallel to the velocity, the object will change direction and move in a curve

The components of C are given by:

Equivalently,

Vector components Review

Vector Components Review

If you know A and B, here is how to find C:

Vector Addition and SubtractionVectors can be resolved into components and the components added separately; then recombine to find the resultant vector.

Projectile Motion

An object moving near the earths surface feels no acceleration in the horizontal direction (forgetting about air resistance)

But it feels gravity in the downwards direction

That is known as projectile motion

Projectile MotionA projectile launched in an arbitrary direction may have initial velocity components in both the horizontal and vertical directions, but its acceleration is always downward.

Projectile MotionThe vertical motion is the same as if the object were thrown straight up or down with the same initial y velocity, and the horizontal velocity is constant.

Projectile Motion

The range of a projectile on level ground is maximum (if there is no air resistance) for a launch angle of 45°.

Projectile Motion

Some thinking problems

• From the same height (and at the same time), one ball is

dropped and another ball is fired horizontally. Which one will

hit the ground first?

• Which ball has the greater velocity at ground level?

• A man on a fast moving train throws a ball straight up – will

he catch it without having to move?

• Draw the path of the man and the ball in space

Example problem

Man throws ball horizontally, with speed 10 m/s, from a balcony 10 meters high.

How far from the building does the ball land?

Example problem

Man throws ball horizontally, with speed 10 m/s, from a balcony 10 meters high.

How far from the building does the ball land? (x-max)

a) Find time to hit the groundb) Calculate horizontal distance

travelled in that time