Vectors and Scalars

Scalar: a value that ONLY contains a magnitude (a ‘size’)

• examples of scalar quantities include distance, speed, and work.

  • They only deal with magnitude and not in what direction the quantity is going

Vector: a value that has magnitude AND direction

• examples include displacement, velocity, acceleration, and forces

  • vector quantities sometimes contain an arrow above their values, hinting that direction is also important with the quantity (but the arrow doesn’t actually show the direction)

Multiplying 2 scalar quantities will create a new scalar quantity

Multiplying a scalar and a vector quantity together will result in a vector quantity

Multiplying 2 vector quantities will create a new vector quantity

Visualizing vectors in 2 dimensions

Vector addition

• Magnitude is specified by the length of the arrow

• Direction is specified by the direction of the arrow

• Vector A + Vector B = Vector C

  • The net shift amount is the sum of Vector A and B

    • NOTE: the sum of two vectors must be greater or equal to the difference of their magnitudes, but smaller or equal to the sum of their magnitudes

      • E.g Vector A= 10 m/s, B=7 m/s, then C must be greater than or equal to 3 OR less than or equal to 17

• Vectors don’t necessarily state a starting point, so even if Vector A and Vector B have different starting points, you can put them ‘head to tails’

  

• In order to do this vector addition, you need to break down the vector into its components

Breaking down any vector into its components

• Vector X is the sum of horizonal component and vertical component

  • break down Vector X into its horizontal and vertical components

• This turns a 2D problem into two separate 1D problems

  

Subtracting Vectors

If asked to subtract a vector, simply switch the arrow direction of the vector and then follow the rules as if its a normal addition problem

Kinematics

Displacement

Displacement is an objects net change in position while the distance refers to the magnitude of the displacement

• Distance = position final- position initial

When referring to the displacement of an object, we must also take direction into account (while distance is scalar and doesn’t require a direction).

Since displacement is the net distance traveled by an object, the displacement value can be less than the total distance taken.

• ex: running on a circular track with starting and stopping at the same location.

  • the distance you ran can be 1 mile but if you stopped in the same place you started, then your displacement is 0 (since there was no net movement in position)

• Displacement= positional final - position initial =net distance plus direction

Velocity

Velocity tells us how fast an object’s position changes.

• average velocity= displacement/time OR change in displacement/change in time

The magnitude of the velocity vector is called the speed

• The sprinter’s average speed= total distance/time= 700/100= 7m/s

• The sprinter’s average velocity= displacement/time= 500/100= 5m/s

Acceleration

Acceleration tells us how fast an object’s velocity changes

• average acceleration= change in velocity/time

NOTE: acceleration tells us how fast velocity changes

• an object’s velocity changes if the speed OR direction changes

• so an object can be accelerating even if the speed is constant

If velocity remains constant, then acceleration is 0 (and can be constantly at 0 until velocity changes)

Deacceleration also constitutes as acceleration (because velocity is changing)

If acceleration points in the same direction as the initial velocity, then the object’s speed is increasing

If acceleration points in the direction opposite to the initial velocity, then the object’s speed is decreasing

• vi= 7m/s, vf= 1m/s, t=3seconds

  • acceleration= (1-7)/3= -2m/s²

    • decreasing acceleration

• vi= -2m/s, vf= -5m/s, t=2seconds

  • acceleration= (-5- -2)/2= -1.5m/s²

    • increasing acceleration

Uniformly Accelerated Motion and Free Fall

Motion in which the object’s acceleration is constant

• Uses the' ‘big 5’ equations, but such equations can be used through other means and all 5 account for missing variables.

  
  

• Most common equations asked for on the MCAT are equations 1, 2, and 3

In terms of free fall (objects falling directly down or being thrown directly up), the big 5 equations can be used, but use the gravitational acceleration constant 9.8m/s² for acceleration (round constant to 10m/s²)

Examples:

1. A particle has an initial velocity of 10m/s and a constant acceleration of 3m/s² in the same direction. How far will the particle travel in 4 seconds?

   • missing final velocity so use equation 2

   • d=vi(t)+1/2at²

   • d=10(4)+1/2(3)(4²)= 64m

2. An object starts from rest and travels in a straight line with a constant acceleration of 4m/s² in the same direction until its final velocity of 20m/s. How far does it travel during this time?

   • missing time but are given vi,a,vf and being asked for d- use equation 3 and rearrange to solve for d

   • vf²= vi² +2ad → d= (vf²-vi²)/2a

   • d= (20²)/(2×4)=50m

3. An object is dropped from a height of 80m. How long will it take to strike the ground?

   • missing final velocity, but given a, d, vi and being asked for t, use equation 2

     • vi=0 m/s because its first at rest before being dropped

   • d=vi(t)+1/2at² → d=1/2at²

   • t= sqrt((2×80/10) = 4s

     • NOTE: a is positive in free fall because we are moving in the same direction as gravity

Projectile Motion

Projectile motion would occur when something is experiencing both horizontal and vertical motion

• can be seen when trajectory of an object follows a curve (due to gravity and being thrown at an angle that’s not directly up or down)

To solve projectile motion problems, we must analyze variables in the x and y directions.

• the key is to analyze each direction separately

  • example: split initial velocity into initial velocity in the x and y directions

    • vx is constant; vy changes due to gravitational acceleration

    • vy=0 at the top of a projectile’s trajectory

• You might need to use trigonometry to solve for variables in the x or y direction and then plug such variables into one of the big 5 equations

Example:

A rock is thrown horizontally, with an initial speed of 10m/s, from the edge of a vertical cliff. It strikes the ground 5s later.

1. How high was the cliff?

   • looking for vertical distance

   • because the rock was thrown horizontally, there is no initial vertical velocity

     • the equation will then look like: vy= 1/2-gt²

       • y=1/2(-10m/s²)(5s)² =-125m

       • height of the cliff is 125m because the equation said it fell from a height of 125m

2. How far from the foot of the cliff does the rock land?

   • looking for horizontal distance

     • use the equation x=vix*t

     • x=(10m/s)(5)= 50m

Motion on an inclined plane

Similar thinking/ processes like projectile motion but instead we break up acceleration into x and y components (or parallel or perpendicular to the inclined plane)

• drawing a free body diagram assists in breaking down acceleration components

• again- this will involve some trig.

Graphical Interpretations relating to time, acceleration, displacement, and velocity