U1: 2D Kinematics

A vector is a quantity that has magnitude and direction

● Displacement, velocity, acceleration, and force are all vectors

● In two dimensions, we specify the direction of a vector using an arrow

○ The length of the arrow is proportional to the vector’s magnitude

The Head-to-Tail method is a graphical way to add vectors

● The tail of the vector is the starting point of the vector

● The head of the vector is the final, pointed end of the arrow

These are the steps to take for the head-to-tail method

● Draw an arrow to represent the first vector using a ruler and protractor

● Draw an arrow to represent the second vector

○ Place the tail of the second vector at the head of the first vector

● If there are more than two vectors, continue this process for each vector to be added

● Draw an arrow from the tail of the first vector to the head of the last vector

○ This is the resultant, or sum, of the other vectors

○ To get the magnitude of the resultant, measure its length with a ruler

○ To get the direction of the resultant, measure the angle it makes with the reference frame using a protractor

Vector subtraction is a straightforward extension of vector addition

● The negative of any vector has the same magnitude but is in the opposite direction

● The subtraction of vector B from vector A is simply the addition of -B to A

We often multiply vectors by a positive scalar

● The magnitude changes, but the direction stays the same

In order to split a vector into two, we determine its perpendicular components

● We can use pythagorean identities to do this

○ 𝐴𝑥 = 𝐴 𝑐𝑜𝑠𝛳

○ 𝐴𝑦 = 𝐴 𝑠𝑖𝑛𝛳

If we are only given the perpendicular components, we can also find the resultant

● 𝐴 = √𝐴2x + 𝐴2𝑦

○ This is just the pythagorean theorem

● 𝛳 = 𝑡𝑎𝑛-1(𝐴𝑦/𝐴𝑥)

Projectile Motion is the motion of an object thrown or projected into the air, subject only to the acceleration of gravity

● The object is called a projectile

● The object’s path is called its trajectory

● We consider air resistance to be negligible

These are equations that are applied to kinematic problems (they are given on your reference sheet)

● 𝑥 = 𝑥0 + 𝑣𝑡

● 𝑣 = (𝑣0 + 𝑣)/2

● 𝑣 = 𝑣0 + 𝑎𝑡

● 𝑥 = 𝑥0 + 𝑣0𝑡 + (1/2)𝑎𝑡2 

● 𝑣2 = 𝑣a20 + 2𝑎(𝑥 − 𝑥0)

These are the steps to solve projectile motion problems

● Resolve or break the motion into horizontal and vertical components along the x and y axes

● Treat the motion as two independent one-dimensional motions, one horizontal and the other vertical. The kinematic equations now look like this:

○ Horizontal: 𝑥 = 𝑥0 + 𝑣𝑡

○ Horizontal: 𝑣𝑥 = 𝑣0x = 𝑣x =velocity is a constant

○ Vertical: 𝑣𝑦 = 𝑣0 − 𝑔𝑡

○ Vertical: 𝑦 = 𝑦0 + .5(𝑣0y + 𝑣y)𝑡

○ Vertical: 𝑦 = 𝑦 + 𝑣0y𝑡 + (1/2)𝑔𝑡2

○ Vertical: 𝑣𝑦2 = 𝑣20y + 2𝑔(𝑦 − 𝑦0)

● Solve for the unknowns in the two separate motions

○ The only common variable between the motions is time

○ The problem solving procedures here are the same as for one-dimensional kinematics

● Recombine the two motions to find the total displacement

○ We do this by using the resultant vectors equations

Velocities in two dimensions are added using the same analytical vector techniques:

● 𝑣𝑥 = 𝑣 𝑐𝑜𝑠𝛳

● 𝑣𝑦 = 𝑣 𝑠𝑖𝑛𝛳

● 𝑣 = √𝑣2x + 𝑣2y  

● 𝛳 = 𝑡𝑎𝑛-1(𝑣𝑦/𝑣𝑥)

Relative Velocity is velocity relative to some reference frame

● Relative velocities are one aspect of relativity, the study of how different observers moving relative to each other measure the same phenomenon

● Classical relativity is limited to situations where speeds are less than about 1% of the speed of light