Physics
Kinematic Equations in 1D
Standard Equations
Summary
Formula | Missing | Quantities Present |
---|---|---|
Deriving Displacement Formulas
Displacement when object moves with constant velocity
Deriving
Displacement when object accelerates from rest
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
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
Symbol | Description |
---|---|
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
Symbol | Description |
---|---|
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
Formula | Missing | Quantities 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