Vectors - advanced Projectile Motion
Horizontal Launch Launch at an Angle
Use vector components to determine displacement in two dimensions Use trigonometry to determine the angles of resultant vectors Calculate trajectories of objects in projectile motion Model and compute centripetal forces Analyze and compute forces and trajectories for objects in uniform circular motion
Vectors can be added graphically by placing them tip-to tail
[Image](image.png) ### Vector addition
Adding vectors in a straight line requires placing the vectors tip-to-tail, then drawing a resultant vector.
The length and direction of the vectors do not change, so the vectors remain unchanged
Resultant vector - a vector that results from the sum of two other vectors
Resultant vectors always point from the first vector’s tail to the last vector’s tip
Walking forward on a moving train
Walking backward on a moving train
Adding vectors that are not parallel allows trajectories and forces to be analyzed in multiple dimensions
When two vectors are added at right angles (90°) to each other, trigonometry can be used to calculate their resultant
Vectors can be visualized over an x-y coordinate plane so they are easier to analyze
Vectors that are not parallel to an x-y axis can be broken into components that are Component - a vector parallel to the x- or y- axis Component vectors make the original vector into a hypotenuse of a right triangle Right triangles allow the use of the Pythagorean theorem to calculate vector lengths
The process of breaking a vector into its components is called vector resolution Any vector can be written as the sum of its components
Vector components can be calculated using trigonometry
Components can accumulate
Vector components can be accumulated, and a final resultant found
Adding y-components gives the y-component of the resultant. Doing the same for the x-components gives the x-component of the resultant.
Trigonometric functions can be used to determine the components
The angle of the resultant from the positive x-axis is the inverse tangent of its components
An airplane takes off at an angle of 4.60° above the horizon and climbs to cruising altitude at an average velocity of 114.3 km/h. Sketch the scenario as described. Label as many components as possible. What is the ground speed of the plane? What is the plane’s rate of climb? How long will the plane take to reach a cruising altitude of 2 000.0 m? How far downrange will the plane be when it reaches 2 000.0 m?
A cargo ship leaves port heading slightly north of east. GPS shows an island that lies 201 km directly east. Shallow reefs require that ships be at least 3.55 km north of the island for safe passage. Sketch the scenario as described. Label as many components as possible. Calculate the ship’s displacement from port as it passes the island. The navigator wants to set a heading for 1° north of east. Will this get them safely past the island’s reefs?
A curious thing about vector components and velocity Time Dilation The speed of light is constant, independent of any observer To an observer on-board a spacecraft, light will bounce between two mirrors at a constant rate To an observer on Earth, the mirrors are moving relative to the beam of light because the second mirror is no longer where it was when the light left the first Because the light has now travelled a longer distance at a constant velocity, we explain this by saying time itself has changed.
Example 3 A model rocket reaches a maximum altitude of 214.6 m. Wind from the east pushed it 48.60 m from the launch pad. Calculate its displacement from the launch pad when it reaches its maximum altitude. At what angle from the horizon would the rocket be visible at maximum altitude? Sketch the situation, label as many vectors and elements as possible.