Post on 30-Jan-2015
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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