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Kinematics is the science of describing the motion of objects using words, diagrams, numbers, graphs, and equations. Kinematics is a branch of mechanics. The goal of any study of kinematics is to develop sophisticated mental models that serve to describe (and ultimately, explain) the motion of real-world objects.
Kinematics is the branch of classical mechanics that describes the motion of points, objects and systems of groups of objects, without reference to the causes of motion (i.e., forces ). The study of kinematics is often referred to as the “geometry of motion.” Objects are in motion all around us.
- Overview
- What are the kinematic formulas?
- What is a freely flying object—i.e., a projectile?
- How do you select and use a kinematic formula?
- How do you derive the first kinematic formula, v=v0+at ?
- How do you derive the second kinematic formula, Δx=(v+v02)t ?
- How do you derive the third kinematic formula, Δx=v0t+12at2 ?
- How do you derive the fourth kinematic formula, v2=v02+2aΔx ?
- What's confusing about the kinematic formulas?
- Example 1: First kinematic formula, v=v0+at
Here are the main equations you can use to analyze situations with constant acceleration.
What are the kinematic formulas?
The kinematic formulas are a set of formulas that relate the five kinematic variables listed below.
ΔxDisplacement
tTime interval
v0 Initial velocity
The kinematic formulas are a set of formulas that relate the five kinematic variables listed below.
ΔxDisplacement
tTime interval
v0 Initial velocity
v Final velocity
a Constant acceleration
It might seem like the fact that the kinematic formulas only work for time intervals of constant acceleration would severely limit the applicability of these formulas. However one of the most common forms of motion, free fall, just happens to be constant acceleration.
All freely flying objects—also called projectiles—on Earth, regardless of their mass, have a constant downward acceleration due to gravity of magnitude g=9.81ms2 .
g=9.81ms2(Magnitude of acceleration due to gravity)
A freely flying object is defined as any object that is accelerating only due to the influence of gravity. We typically assume the effect of air resistance is small enough to ignore, which means any object that is dropped, thrown, or otherwise flying freely through the air is typically assumed to be a freely flying projectile with a constant downward acceleration of magnitude g=9.81ms2 .
This is both strange and lucky if we think about it. It's strange since this means that a large boulder will accelerate downwards with the same acceleration as a small pebble, and if dropped from the same height, they would strike the ground at the same time.
[How can this be so?]
We choose the kinematic formula that includes both the unknown variable we're looking for and three of the kinematic variables we already know. This way, we can solve for the unknown we want to find, which will be the only unknown in the formula.
For instance, say we knew a book on the ground was kicked forward with an initial velocity of v0=5 m/s , after which it took a time interval t=3 s for the book to slide a displacement of Δx=8 m . We could use the kinematic formula Δx=v0t+12at2 to algebraically solve for the unknown acceleration a of the book—assuming the acceleration was constant—since we know every other variable in the formula besides a —Δx,v0,t .
Problem solving tip: Note that each kinematic formula is missing one of the five kinematic variables—Δx,t,v0,v,a .
1.v=v0+at(This formula is missing Δx.)
2.Δx=(v+v02)t(This formula is missing a.)
3.Δx=v0t+12at2(This formula is missing v.)
This kinematic formula is probably the easiest to derive since it is really just a rearranged version of the definition of acceleration. We can start with the definition of acceleration,
a=ΔvΔt
[Isn't this the average acceleration?]
Now we can replace Δv with the definition of change in velocity v−v0 .
a=v−v0Δt
Finally if we just solve for v we get
A cool way to visually derive this kinematic formula is by considering the velocity graph for an object with constant acceleration—in other words, a constant slope—and starts with initial velocity v0 as seen in the graph below.
The area under any velocity graph gives the displacement Δx . So, the area under this velocity graph will be the displacement Δx of the object.
Δx= total area
We can conveniently break this area into a blue rectangle and a red triangle as seen in the graph above.
The height of the blue rectangle is v0 and the width is t , so the area of the blue rectangle is v0t .
The base of the red triangle is t and the height is v−v0 , so the area of the red triangle is 12t(v−v0) .
There are a couple ways to derive the equation Δx=v0t+12at2 . There's a cool geometric derivation and a less exciting plugging-and-chugging derivation. We'll do the cool geometric derivation first.
Consider an object that starts with a velocity v0 and maintains constant acceleration to a final velocity of v as seen in the graph below.
Since the area under a velocity graph gives the displacement Δx , each term on the right hand side of the formula Δx=v0t+12at2 represents an area in the graph above.
The term v0t represents the area of the blue rectangle since Arectangle=hw .
The term 12at2 represents the area of the red triangle since Atriangle=12bh .
[Wait, how?]
To derive the fourth kinematic formula, we'll start with the second kinematic formula:
Δx=(v+v02)t
We want to eliminate the time t from this formula. To do this, we'll solve the first kinematic formula, v=v0+at , for time to get t=v−v0a . If we plug this expression for time t into the second kinematic formula we'll get
Δx=(v+v02)(v−v0a)
Multiplying the fractions on the right hand side gives
Δx=(v2−v022a)
People often forget that the kinematic formulas are only true assuming the acceleration is constant during the time interval considered.
Sometimes a known variable will not be explicitly given in a problem, but rather implied with codewords. For instance, "starts from rest" means v0=0 , "dropped" often means v0=0 , and "comes to a stop" means v=0 . Also, the magnitude of the acceleration due to gravity on all freely flying projectiles is assumed to be g=9.81ms2 , so this acceleration will usually not be given explicitly in a problem but will just be implied for a freely flying object.
People forget that all the kinematic variables—Δx,vo,v,a —except for t can be negative. A missing negative sign is a very common source of error. If upward is assumed to be positive, then the acceleration due to gravity for a freely flying object must be negative: ag=−9.81ms2 .
The third kinematic formula, Δx=v0t+12at2 , might require the use of the quadratic formula, see solved example 3 below.
A water balloon filled with Kool-Aid is dropped from the top of a very tall building.
What is the velocity of the water balloon after falling for t=2.35 s ?
Assuming upward is the positive direction, our known variables are
v0=0 (Since the water balloon was dropped, it started at rest.)
t=2.35 s (This is the time interval after which we want to find the velocity.)
ag=−9.81ms2 (This is implied since the water balloon is a freely falling object.)
Kinematics is a subfield of physics and mathematics, developed in classical mechanics, that describes the motion of points, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause them to move.
Week 1: Kinematics. Week 1: Introduction; Lesson 1: 1D Kinematics - Position and Velocity. 1.1 Coordinate Systems and Unit Vectors in 1D Position Vector in 1D; 1.2 Position Vector in 1D; 1.3 Displacement Vector in 1D; 1.4 Average Velocity in 1D; 1.5 Instantaneous Velocity in 1D; 1.6 Derivatives; 1.7 Worked Example - Derivatives in Kinematics
About this unit. Having a specific understanding of an object's position, acceleration, velocity, and motion comes in handy in situations ranging from bobsledding to launching rockets into outer space. Let's explore the concepts and equations that govern how objects move, and learn how to calculate the specifics of an object's motion.
Ciencia. Física avanzada 1 (AP Physics 1) Unidad 1: Cinemática e introducción a la dinámica. 500 posibles puntos de dominio. Dominado. Competente. Familiar. Intentado. Sin empezar. Cuestionario. Prueba de unidad. Acerca de esta unidad. En esta unidad se exploran los conceptos y las ecuaciones que gobiernan cómo se mueven los objetos.
