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3.1 Kinematics in Two Dimensions: An Introduction
Explain the properties of a projectile, such as gravity, range, maximum height, and trajectory.
The principle of independence of motion can be applied to solve projectile motion problems.
Relative velocity is determined by the principles of vector addition.
The arcs of a basketball, a satellite, a bicycle, a swimmer, and a puppy are a few examples of motions along curved paths.
Most motions in nature follow curved paths.
The motion of a ball on a pool table or a skater on an ice rink is described by two-dimensional kinematics.
A car following a winding mountain road is described by three-dimensional kinematics.
The two- and three-dimensional kinematics are extensions of the one-dimensional kinematics that were developed for straight-line motion.
This simple extension will allow us to apply physics to many more situations, and it will also give us unexpected insights about nature.
In a city like New York, it's rare for walkers and drivers to travel in straight lines.
They must follow the roads and sidewalks.
The blocks are the same size in this scene.
You are forced to take a two-dimensional path because the straight-line path that a helicopter might fly is blocked to you as a pedestrian.
You walk 14 blocks, 9 east and 5 north.
The shortest distance between two points is a straight line, according to an old adage.
The Pythagorean theorem can be used to find the straight-line distance because the two legs of the trip form a right triangle.
The hypotenuse is related to the length of the legs of a right triangle.
The relationship is given by something.
The length of the triangle in units of city blocks is considerably shorter than the 14 blocks you walked.
It appears that "9" and "5" have only one digit.
The 14 blocks walked by the pedestrian are larger than the straight-line path followed by a helicopter.
The blocks are the same size.
We use arrows to represent the one-dimensional kinematics.
The length of the arrow is related to the magnitude of the object.
The straight-line path is given by the horizontal and vertical components of the motion.
The first shows a displacement east.
The second is a displacement north.
The third vector has a total displacement of more than 10 blocks.
There is a straight-line path between the two points.
The right triangle is formed by the vectors that are perpendicular to each other.
The magnitude of the total displacement can be calculated using the Pythagorean theorem.
His or her motion eastward affects how far he or she walks.
How far he or she walks north is dependent on his or her motion northward.
The horizontal and vertical parts of motion are not related.
Motion in the horizontal direction does not affect motion in the vertical direction.
This is true if you walk in one direction first, followed by another.
It's true of more complicated motion that involves movement in two directions at the same time.
Let's compare the motions of two baseballs.
A baseball is dropped.
One is thrown from the same height and the other follows a curved path.
The balls are captured by a stroboscope as they fall.
The motions of two identical balls are shown in this picture.
Each position has an equal time interval.
The horizontal and vertical velocities are represented by the arrows.
The ball on the right has a higher initial horizontal velocity than the ball on the left.
The vertical velocities and positions are the same for both balls.
The motions of the vertical and horizontal are independent.
The vertical positions of the two balls are the same for each flash of the strobe.
The similarity shows that the vertical motion is not dependent on whether the ball is moving horizontally or not.
Careful examination of the ball shows that it travels the same distance between flashes.
There are no additional forces on the ball in the horizontal direction after it is thrown.
The result means that the horizontal velocity is unaffected by either vertical motion or gravity.
The case is only true for ideal conditions.
Air resistance will affect the speed of the balls.
There are two independent one-dimensional motions in the path of the ball.
The key to analyzing projectile motion is to break it into motions.
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