• 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

3.2 - Addition of Vectors - Graphical Methods

• 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

3.3 - Subtraction of Vectors, and Multiplication of a Vector by a Scalar

• 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

3.4 - Adding Vectors by Components

• 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

Adding Vectors

• 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)

3.5 - Projectile Motion

• 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

3.6 - Solving Projectile Motion Problems

• 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

3.7 - Projectile Motion is Parabolic

• 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

3.8 - Relative Velocity

• 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