Kinematics

Objective

As a result of this lesson you will be able to

  • Define the concepts of motion
  • Describe straight line motion in terms of instantaneous acceleration, average acceleration, average velocity, and instantaneous velocity.
  • Recall the equations of motion
  • Solve problems involving motion with constant acceleration by using the equations of motion
  • Interpret motion graphs and describe their connection to each other
  • Predict the motion of projectiles and correctly seperate its motion into two independent motions
  • Calculate the position, velocity, and acceleration given information
  • Describe how calculus plays a role in kinematics and what the use of derivatives and integrals have on the equations of motion

Prerequisites

This topic expects you to have an understanding of the following in order to successfully learn all the material.

      • A basic understanding of how to solve algebraic expressions
      • A basic understanding of trigonometry to solve for angles
      • Ability to break a vector into its components
      • An understanding of derivatives in calculus to understand calculus section.

You can find many of these topics covered here for your convenience.

Use this knowledge check to see if you already know many of the concept questions involving kinematics. If you are able to pass this test then you can move on to the next topic. If you do not do well then review the sections that the knowledge check identifies as areas requiring reinforcement.

Knowledge Check

This is a pre quiz to give you an idea of any areas you may need to have additional review.

Question

A runner is on the last mile of his run. He decides to slow his pace as he cools down. Is it correct to say that the runner is accelerating as he slows down?

Introduction

In Kinematics the main thing we are concerned with is how we describe the motion of a particular object. Usually we will be presented with only parts of an objects description of motion (velocity, position, acceleration) and will be required to determine the missing information. Kinematics problems are a lot like solving puzzles. You have to fit the pieces together in order to see the whole picture. It is our job to determine what pieces we have and what pieces we need to solve. For these problems we will only consider situations in which motion has either a constant velocity or a constant acceleration.
kinematics Chart

Connection

You should already have the math skills to solve basic algebra problems and a knowledge of trigonometric identities. One application of mathematics is to help determine the position of objects. When given an equation that describes the position of an object, you can find the velocity and acceleration from the same equation.

Role

Kinematics is used to describe the motion of objects. The equations learned in this section will allow you to predict the path of an object as well as its velocity and acceleration. You will usually be given a piece of information that relates to the object's position, velocity, or acceleration and be asked to find the missing information that describes the objects motion. Kinematics is only concerned with describing motion and it does not care about what is causing motion to occur. It is very important to be able to give an accurate description of an object. This will be our first step to understanding the branch of physics known as mechanics.








Displacement and Distance

 Displacement  is an object’s change in position. Displacement can also be viewed as a vector. If you connect the points from an object’s initial position to its final position then the length of the vector is the displacement. If you create a vector to represent the displacement of an object then you will notice that it does not depend on the path that the object took to get to that location. Displacement is only concerned with the initial and final positions. Displacement is a scalar quantity meaning it carries no information in regard to direction. Be careful if you decide to view displacement as a vector since you must remember that displacement is a scalar. Displacement is usually viewed as and if you know it will only move in one direction you can label that as an x direction and use for the y direction.

Equation/Definition

Distance refers to how much ground has been covered by an object. Displacement refers to how far from the original position the object is.

    \[displacement=\Delta s\]

    \[distance = total\:path\:length\]

distance and displacement

In case you still may be confused between the two terms here is another representation about displacement and distance.

Example

A rock is thrown straight up 10m and then comes back down and is caught. What is the displacement of the rock that was thrown?

Solution
Identify
Setup
Execute
Evaluate








Speed and Velocity

We use speed and velocity to describe the motion of an object. These are sometimes confused with each other and need to be differentiated. Speed does tell us how fast an object is moving and so does velocity. What is important though is that velocity also gives us the direction.

Equation/Definition

Speed is the distance traveled by an object over a time interval. Velocity gives both speed and direction. This means that velocity is the displacement of an object over time with an indicated direction.
average\:speed=\frac{total\:distance}{time}
average\:velocity=\overline{\textbf{v}}=\frac{displacement}{time}=\frac{\Delta \textbf{s}}{\Delta t}

Example

If a runner completes a race in 2 minutes and 10 seconds, what is their average speed and magnitude of their average velocity? The racetrack is 500m and report your answer in m/s.

Solution
Identify
Setup
Execute
Evaluate







Acceleration

Acceleration is used to describe the change in velocity. Acceleration is a vector so it has a magnitude and direction. Acceleration only occurs when an object's velocity is changing. This means that if an object is moving with a constant velocity it has zero acceleration. This can be tricky because you may think there is an acceleration since the object is moving but remember acceleration is a description of the rate at which velocity is changing. On earth we experience an acceleration from gravity that is measured at 9.8\:\frac{m}{s^{2}}.

Equation/Definition

Acceleration measures the rate of change of an object's velocity.
average\:acceleration=\overline{\textbf{a}}=\frac{change\:in\:velocity}{time}=\frac{\Delta \textbf{v}}{\Delta t}


Example

A car is driving on the freeway at a constant speed of 75 miles per hour for 1 minute. What is the acceleration of the car during this time?

Solution
Identify
Setup
Execute
Evaluate








Uniformly-Accelerated Motion and the Big Five

The common type of motion that will be studied consists of motion with a constant acceleration(can be zero). We will typically want to break up motion in multiple directions into smaller problems. The way we can do this is to analyze the motion in a single direction at a time. Motion in 3D is simply made up of the motion in each of the three directions. This can be a simple technique to understand complex motion. When we deal with motion in a single dimension (x, y, z for example) we can use the + and - sign to indicate direction and do not need to pay too much attention to vector quantities. The three main quantities that we will have to remember when studying motion are displacement, velocity, acceleration, and time. Each of these can have an initial and final quantity and we will have to determine what is missing in order to know which equation to use to determine the others.

Equation/Definition

Big 5 Kinematics Equations
Number Equation Missing Variable
1 \Delta s=v\Delta t a
2 \Delta v=a\Delta t \Delta s
3 \Delta s= v_{0}\Delta t + \frac{1}{2}a(\Delta t)^{2} v
4 \Delta s= v\Delta t - \frac{1}{2}a(\Delta t)^{2} v_0
5 v^{2}=v_{0}^{2}+2a\Delta s \Delta t

Example

An object has an initial velocity of 10 m/s in a straight line. Five seconds later its velocity is 20 m/s. How far did the object travel during this time interval?

Solution
Identify
Setup
Execute
Evaluate








Kinematics with Graphs

Another method to solve kinematics equations is through the use of graphs. Graphs are a way to represent the motion of objects instead of using algebraic equations. You will typically be given graphs that show position vs time, velocity vs time, and acceleration vs time. Take a look at this animation below to get an understanding of how position, velocity, and acceleration are related.

If you remember the equations from above of what average velocity was and average acceleration, then you can see that the graphs are simply the slope of the graph to the left of it. For a reminder here are the equations

    \[average\:velocity==\frac{displacement}{time}=\frac{\Delta \textbf{s}}{\Delta t}\]

    \[average\:acceleration=\frac{change\:in\:velocity}{time}=\frac{\Delta \textbf{v}}{\Delta t}\]

Again, velocity is the slope of position-vs-time graph and acceleration is slope of velocity-vs-time graph. If you know a bit of calculus then the derivative of a function will give you the rate of change of that function....AKA the slope. So we use derivatives to find the slope of functions at any point along the graph.
Another important piece of information to get from graphs is not only the rate of change of another graph but the displacement. Remember we graph position, velocity, and acceleration versus time. If you were to find the area between the graph and time axis you would find information about another description of an object's motion. The area determined in the acceleration graph gives the net change of velocity. The area determined in the velocity graph gives the net change of position.





Relating Graphs
  1. What Slopes tell us
    • Slope of position-vs-time graph gives the velocity
    • Slope of velocity-vs-time graph gives the acceleration
  2. What Areas tell us
    • Area between the velocity-vs-time graph and t axis equals the object's displacement
    • Area between the acceleration-vs-time graph and t axis equals the object's change in velocity




Take a look at the following activity to see how position, velocity, and acceleration are related.

The Moving Man

Click to Run
Activity Questions
Construct a velocity-time graph that would depict following actions. The man walking backwards for a few seconds. He then stands still for another few seconds and then finally walks forward twice as fast for a few seconds.(you can make a few seconds 5 if you wish)
In the above problem the man walked backwards for a few seconds and he also walked forwards for a few seconds at twice the speed. If the initial speed the man began to walk backwards with was doubled, what would happen to the amount of time it takes to stop? You may want to refer to the activity or any equations you know to think about what would happen.
Draw a position-time graph of a car that is moving forward and then applies the breaks to come to a complete stop.
Are you able to describe a scenario that would result in the following motion graphs? You can use the activity to give you some hints about how to create common line segments.
Does an increase in speed mean that acceleration is positive?






Free Fall

One common example that kinematics is used for is for objects that are in free fall. This means that an object is under a constant acceleration(air resistance neglected) due to gravity as it falls from a height above the ground. Since we are only looking at how gravity is affecting the fall then the object is considered to fall freely. Gravity has a value of 9.8\frac{m}{s^{2}} directed toward the center of the earth but you can round it to 10\frac{m}{s^{2}} for an approximation within this site. Remember, I said that gravity is directed toward the center of the earth so you will have to make sure when you choose your coordinate system if g will be positive or negative when you use the big five kinematics equations from above.

>Take a look at this simulation below showing two balls falling. This is a prelude to projectiles.

Notice how even though one ball is moving toward the right, they both fall down at the same rate. Gravity is the only thing affecting each ball equally. What you are seeing is that the motion the ball experiences horizontally(left and right) does not affect the motion vertically (up and down). The horizontal and vertical motions are independent of each other.

Equation/Definition

Free fall is any motion that is the result of its weight due to gravity. On earth we use a common value of 9.8 \frac{m}{s^{2}} for the acceleration an object would experience while falling.

Example

How long does it take for a feather to hit the ground if it is dropped from a cliff that is 100 m high?(remember that we are neglecting air resistance)

Solution
Identify
Setup
Execute
Evaluate








Projectile Motion

Projectile motion involves kinematics equations in two directions. We will typically ignore air resistance and the projectile will be launched at an angle while only experiencing the force of gravity during its travel. Some of the common questions that will be asked about this launched projectile will be its time of flight, range, maximum height, angle of launch, and possibly additional kinematic descriptions of its motion(velocity, acceleration, position).

Equation/Definition

Projectile motion is the continuous motion an object experiences after being launched or dropped. Once the object is in motion it continues its motion due to its inertia and influence of gravity. The path of travel is parabolic.

Take a look at this activity to learn more about projectile motion. Can you hit the target with the activity below?

Once you have experimented with the activity try to test your understanding with the related questions below.

Activity Questions
Set the following conditions for the a car being shot from the cannon and answer the coresponding questions
Set the following conditions for two different objects and determine if the results from the previous questions change.
Now try one more time to set the same conditions as before but with two different objects and make sure to add air resistance. Will the results change now?








Calculus Note

So far we have been performing physics without the use of calculus. Calculus concepts such as integrals and derivatives are a powerful tool to inspect mathematical equations and extract information from them. If you understand the concepts of calculus you can also use your understanding to more accurately construct motion graphs from objects. This section will briefly use the concepts of calculus on some problems but if you do not know how to perform simple integrals or derivatives then you should check out the math section of BnF Scribbles for a more in depth coverage of the topic.

Teaser

Can you answer this correctly with your knowledge of kinematics? Think about why the outcome of the simulation occurs. What is causing the balls to accelerate? Where does this acceleration occur? This will be revisited in another section for further investigation so do not be too upset if you cannot understand what is happening.






Key Terms






Review Questions

  1. A geologist can study the shock waves produced from earth quakes. The two types produced are P-waves (pressure/primary) and S-waves (secondary/shear). P-waves can travel at 6.5 km/s and S-waves can travel at 3.5 km/s. This speed can change depending on the material that the waves travel through. By recording the time between each wave is detected, a geologist can estimate the distance from which an earthquake came from. If the time difference recorded was 45 seconds, how far away was the earthquake?
  2. It is common to experience traffic when driving through the city freeways during rush hour. On a good day you are able to drive at a speed of 60 mph but with traffic you can only drive 40 mph. If your trip normally takes 2 hours how much long does it take on a day with traffic?
  3. You just a new pair of Nike shoes and want to go for a run. You decide to run 100m north at 5m/s and then 200m east. What would your average speed and average velocity be for the path you run?
  4. Explain how if an object that has a constant acceleration is able to reverse its direction of travel. Could this occur twice?
  5. Identify reasons why instantaneous velocity and average velocity are different and give an example of how they can be the same.
  6. On a boring rainy day you sit inside and stare at the rain as it falls outside. You then think about the rain drops as they fall from the clouds. If a raindrop falls every second from a cloud, does the distance between them increase as it falls to the ground?

Solutions

Leave a Reply

Your email address will not be published. Required fields are marked *