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Lesson 6: Two Dimensional Motion


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Download here: Ontario Curriculum Expectations

Human subtlety will never devise an invention more beautiful, more simple or more direct than does Nature, because in her inventions, nothing is lacking and nothing is superfluous.

Leonardo DaVinci

When objects are thrown, they tend to follow a curved parabolic path called a trajectory. This is the result of two independent motions:

  1. A vertical component that is affected by gravity.
  2. A horizontal component that is affected only by air resistance and which we will ignore as it has minimal effect at low velocities.
This is an image of a football player throwing a football. The path of the football is shown.

The red line traces the path, or trajectory, of the football. This trajectory follows the path of a parabola. The football is moving in two dimensions:up and then down; and2. horizontally to the right.

Consider the vertical motion of the cannonball; as it moves upwards it is slowed by gravity until it stops moving upwards. It is then accelerated downwards by gravity, falling faster and faster as it approaches the ground.Consider the horizontal motion of the cannonball; it continues at a constant speed as it moves to the right.

This is an image of a cannon firing a cannon ball from the edge of a cliff. The trajectory of the cannon ball is shown.

In the diagram shown below, an orange ball and a green ball are dropped at the same time. However, the green ball is given a sideways push just as it is dropped. The time interval between any two adjacent lines is the same.

This image shows an orange ball falling straight down and a green ball that is moving horizontally as it is falling.
Vertical MotionHorizontal Motion
As the balls fall, they do so at the same rate, covering larger and larger vertical distances in each successive interval due to their acceleration due to gravity.The green ball is seen to cover the same horizontal distance over each successive time interval indicating that it is moving at a constant speed horizontally and toward the right.

This shows that the horizontal motion of the ball has no effect on its vertical motion.

When solving two dimensional problems, list all of the vertical variables separately from the horizontal variables. You can then solve the vertical and horizontal components of the problem independently.

Example

While moving out of her 30.0 m high dorm room apartment building, a student throws her suitcase full of clothes horizontally out of a window at a speed of 8.0 m/s.

A) How long is will it take the suitcase to hit the ground below?

B) How far from the building will the suitcase land?

This is a diagram showing the trajectory of the suitcase that has been thrown out of the window.

The vertical component of the problem (part A) must be solved separately from the horizontal part of the problem (part B). The only variable that will be the same in both parts is the time. The suitcase will be moving vertically for the same amount of time that it is moving horizontally.

Part A: Vertical MotionPart B: Horizontal Motion
Up is positive, down is negative.Given:equationequationequation, remember that this is the initial vertical velocity.Required:Ī”t = ?Solution:equation Since equation, we can eliminate the equation term.equationequationequationIt will take the suitcase 2.47 s to hit the ground.Right is positive, left is negative.Given:equationThis was obtained from part A.equationequationsince there is no acceleration in the horizontal directionRequired:equationSolution:equationequationequationThe suitcase will land 19.8 m away from the building.

Watch Shoot the Monkey to see an example of projectile motion.