Physics

Kinematic Equations in 1D

Conventions

Graphical Representation

Standard Equations

Summary

FormulaMissingQuantities Present

Basics

Constant Velocity
Uniform acceleration

Miscellaneous

Deriving Displacement Formulas

Displacement when object moves with constant velocity

Deriving

Displacement when object accelerates from rest

Deriving

Displacement when object accelerates with initial velocity

Deriving

Deriving The Other Kinematic Formulas

Deriving

Given

We can rearrange from equation (1) like so

Therefore

Deriving

Given

from equation (1) can be rearranged as

from equation (2) can be rearranged like so

Using the following equations from above

  • from equation (3)
  • from equation (4)

Rearranging equation (5)

Rearrange again to obtain the more common form

TODO

Two-dimensional Projectile Motion

Conventions

Summary

It's easy to see in the above visualization that and increase linearly, while is non-linear.

Formulas

Displacement & Projectile Position

Generalized

In general (without respect to any or axis values)

Where the distance traveled or displaced is

In terms of and axis values
With respect to the axis

The displacement of a given projectile in terms of the axis is

Since

Which can be read as (in terms of the axis)

With respect to the axis

The displacement of a given projectile in terms of the axis is

Note that (because there is no force acting on the projectile in the horizontal direction), and therefore the initial and final velocities are the same. I.e. it's constant throughout. Therefore in summary

  • and therefore we will simple refer to the velocity vector as as .

Therefore we can simplify equation (1) considerably

Solving Projectile Motion Problems

Projectile Motion

In terms of the axis

TODO

TODO

In terms of the axis

TODO

TODO

In Summary

Initial Quantities

Derived expressions

Solutions

Projectile Motion from an initial height, with given initial velocity and angle

Given

  • A projectile angle
  • The initial height
  • The initial velocity

We can therefore derive the the initial velocities for and in terms of the given angle and initial velocity.

Given the general formulas for displacement and velocity

Which this information, we will derive specific equations in terms of the and axes governing the projectile.

In terms of the axis
Deriving displacement as a function of time

Using the general formula from above in terms of as a function of time.

Which we can simplify using the following facts

  • From the given depiction of the problem, we know that .
  • There is no acceleration along the axis, so .
  • as shown above.

Therefore

Deriving velocity
In terms of the axis
Deriving displacement as a function of time

Using the general formula from above in terms of as a function of time.

Which we can simplify using the following facts

  • Initial height is given to us which we will represent as , for the sake of generality.
  • Acceleration along the axis is the constant for gravity, so .
  • as shown above.

Therefore

Deriving velocity

In summary

To find the range

We know that at the moment of impact , therefore we can use equation

Rearranging a bit and setting , we can see that solving for will yield the time at which .

Therefore

Plugging the solution for (and ignoring the negative or non-real solutions for ) into will yield the horizontal displacement (range) at the time . Therefore:

To find the maximum vertical displacement (i.e. peak height)

We begin with equation

We know that at the moment our projectile crests its trajectory, the vertical component of our projectile will be zero. Therefore . To find the time, we simply solve for .

Therefore, knowing the time at which our projectile crests its trajectory, we simply plugin our solution for into the function given in equation . I.e:

To find the velocity at a given moment of time

Given some time which we will denote as , to find the velocity we simply plug in our given values for and into equations and . I.e.

With the given value for , yielding the vector at time , which we will denote as

To define the vector in terms of engineering notation, (i.e.

To define the vector in terms of magnitude (which we will denote as ) and direction (which we will denote as

Range

The distance a projectile travels is called its range.

Only applies in situations where the projectile lands at the same elevation from which it was fired.

Reasoning About Projectile Motion

Notes

  • An object is in free fall when the only force acting on it is the force of gravity.

Question

Based on the figure, for which trajectory was the object in the air for the greatest amount of time?

Answer

Trajectory A

Explanation

All that matters is the vertical height of the trajectory, which is based on the component of the initial velocity in the vertical direction (). The higher the trajectory, the more time the object will be in the air, regardless of the object's range or horizontal velocity.

Problems

The function in this graph represents an object that is speeding up, or accelerating at a constant rate.

When you throw a ball directly upward, what is true about its acceleration after the ball has left your hand?

Answer: The ball’s acceleration is always directed downward.

Wrong: The ball’s acceleration is always directed downward, except at the top of the motion, where the acceleration is zero.

Question
As an object moves in the x-y plane, which statement is true about the object’s instantaneous velocity at a given moment?
Answer
The instantaneous velocity is tangent to the object’s path
Wrong
  • The instantaneous velocity is perpendicular to the object’s path.
  • The instantaneous velocity can point in any direction, independent of the object’s path.
Explanation
As an object moves in the x-y plane the instantaneous velocity is tangent to the object‘s path at a given moment. This is because the displacement vector during an infinitesimally small time interval is always directed along the object’s path and the velocity vector always has the same direction as the displacement vector.

Relative Motion

Galilean transformation of velocity

The velocity of some object P as seen from a stationary frame must be the sum of and

Where

SymbolDescription
Velocity as measured in a stationary frame
Velocity of an object measured in the moving frame relative to the moving frame
velocity of the moving frame - with respect to the stationary frame

Galilean transformation of velocity (alternate notation)

Given two reference frames \text{ A } and and some object . The velocity of the object can be defined in terms of or as shown

SymbolDescription
The velocity of relative to
The velocity of relative to
The velocity of relative to
The velocity of relative to . It locates the origin of relative to the origin of .

Therefore

Rotational Motion & Kinematics

Basics

Auxiliary Formula Reference

FormulaMissingQuantities Present

A particle moves with uniform circular motion if and only if its angular velocity V is constant and unchanging.

Uniform Circular Motion

Uniform means content speed

Forces and Newton's laws of motion

Newton's laws of motion

Normal force and contact force

Balanced and unbalanced forces

Inclined planes and friction

Tension