Chapter 3 - Kinematics in Two Dimensions; Vectors

Motion of objects is usually considered in multiple dimensions, one such example is projectile motion where objects are projected outwards near Earth’s surface

3.1 - Vectors and Scalars

Vector - A quantity with direction and magnitude
Scalar - A quantity with only magnitude
Vectors are represented by arrows in diagrams modelling problems
- Example - A car’s velocity as it changes may be represented by an arrow whose length represents the magnitude of velocity
Vector quantities are written in boldface with a small arrow, scalars are written in italics
- Vector for velocity: ~v
- Scalar for speed: v

Vector additions is more tricky since direction must be added as well
Tail to Tip Method - Drawing the tail of one vector on the tip of the other, the resultant vector is
To add vectors in direction perpendicular to each other, the Pythagorean theorem is used by treating the two vectors as sides a and b and their sum as side c
The direction of the sum is determined uring trigonometry
- Example - A car moves 30 km east and 40km north, what is the resultant vector of its displacement?
- (30 km)^2 + (40 km)^2 = 2500 km^2
- sqrt(2500 km^2) = 50 km displaced
- arctan(40 km/30 km) = 53 degrees
- Therefore, the car moved 50 km at 53 degrees north of east

The negative of a vector ~v has the same magnitude but opposite direction
Subtracting a vector from another has the same effect as adding its negative
Multiplying a vector by a scalar increases its magnitude by the factor of the scalar

Adding vectors by components is much more accurate and applicable in multiple dimensions Components
A vector ~v on a plane is the sum of two smaller component vectors, one on each axis
To determine the magnitude of each component vector is known as resolving it into its components
Trigonometry can be used to resolve vectors, pretend the vector is the hypotenuse of a right triangle
Sine - The sine of an angle of a right triangle is opposite side/hypotenuse
Cosine - The cosine of an angle of a right triangle is adjacent side/hypotenuse
Tangent - The tangent of an angle of a right triangle is opposite side/adjacent side
If an angle and a component vector are known, trig can solve for the other component vector
If the component vectors are known, inverse trig can be used to solve for the angle

To add vectors using components, resolve each one into its components, add the x and y components individually, and combine the resultant components
Equations used
- vRX = v1x + v2x
- vRY = v1y + v2y
- vR = sqrt(vRX^2 + vRY^2)

Objects moving in the air near Earth’s surface are projectiles, their motion is described by projectile motion
In many cases we do not consider air resistance as its effect is minimal
An object in projectile motion maintains constant velocity in the x direction but accelerates negatively in the y direction
Displacement in the x direction is given by ~dx = (vxi)t
Displacement in the y direction is given by ~dy = −1/2 gt^2
After a given amount of time, the displacements are calculated and added vertically to determine overall displacement

Equations for projectile motion:
- Horizontal motion
- ~vx = vx0
- x = x0 + vx0t
- Vertical motion
- vy = vy0 − gt
- y = y0 + vy0 − 1/2gt^2
- vy^2 = vvy0^2 - 2g(y - y0)
Equations for magnitude of initial velocity based off of angle of launch
- Horizontal velocity
- vx0 = v0cosθ
- Vertical velocity
- vy0 = v0sinθ
Equation for determining range of a projectile (only if yf = y0)
- R = (v0^2 * sin2θ0)/g
- Where θ0 is the angle of launch

Simplifying projectile motion by ignoring air resistance, it is parabolic, or, a projectile moves in a parabola
The basic form of a parabola is y = Ax + Bx2, where A and B are constants, which is very similar to the equation for vertical displacement in projectile motion

Relative velocity is the sum of the vector velocities acting on an object from a frame of reference
- Example - If Car A is travelling 75 km/h and Car B is travelling 100 km/h , the relative velocity of Car
B to Car A is 100 km/h - 75 km/h = 25 km/h
If the velocities are in two different directions, then they can be added/subtracted like any vector
The velocity of object A relative to object B is the opposite of the velocity of object b relative to object A, represented by vBA = -vAB

Note

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