Crossfire off South Shore - BOCA race 2015. Yacht navigation requires a good understanding of velocity, distance and time.
1.2 - Describing Motion
Objectives:
- To know the five variables that describe motion
- To fully appreciate their physical meaning
- To be able to measure all of these variables experimentally
- To understand the importance of visualising the problem before starting to try to solve it.
PROBLEM: You are going on a long bike ride. You ride for one hour at \(5\,\text{km/hr}\), then three hours at \(4\,\text{km/hr}\), and then two hours at \(7\,\text{km/hr}\). How far did you go? No equations allowed.
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Measuring Speed
An instantaneous speed is a measure of how fast something is moving at a particular moment in time. An average speed has a value which is calculated over a period of time whether a nanosecond or a million years. In local terms: instantaneous speed is what the police catch you going with their radar gun, whereas the average speed is what your mother tells you off for riding when you get home from town in less time than she does! |
Speed traps record your speed at a single moment in time - instantaneous speed.
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Consider a \(100 \,\text{m}\) sprint, during the start of the race the speed of the runners will increase until it reaches a maximum. All sprinters train to increase their maximum speeds and to be able to reach their maximum speeds in the shortest possible time. After five seconds the athlete may have covered \(40\, \text{m}\). Their average speed over this total distance is \(8 \,\text{m/s}\). Yet their speed at that moment of time may be at its maximum say \(12\,\text{m/s}\). In practical situations instantaneous speeds cannot easily be calculated, as that would require the measurement of an infinitesimally small time. The best we can do is to find the average speed over a small time interval. A speedometer in a car will display the average speed of the vehicle over a period of approximately \(0.5 \,\text{s}\) - and, no this is not getting you off that speeding ticket!
As with earlier courses, speed is measured in the lab using a variety of techniques - the most accurate being the light gate. This measures the time that it takes for an interrupt card to pass through the infrared beam. Strictly speaking it measures the average speed of the card. The shorter the card is, the more accurate the measurement will be, with the limit being the thickness of the beam itself. Other methods involve fixed time intervals - of which the best technique is careful frame-by-frame analysis of video footage or GPS position data. GPS units also calculate the average speed over the short intervals between fixes from the satellite signals.
Velocity is the vector version of speed. It has a direction as well as a magnitude. The speedometer on a bike does not take into account the direction that you ride in. The vector distance (displacement) from Spanish Point to Dockyard is pretty short - about \(3\,\text{km}\). However, the distance to ride between the two is much longer - nearly \(27\,\text{km}\)! For simplicity, assume that it takes \(1.0\,\text{hr}\) to travel the distance. The average speed would be \(27\,\text{km/hr}\), whereas the velocity would be only \(3\,\text{km/hr}\).
As with earlier courses, speed is measured in the lab using a variety of techniques - the most accurate being the light gate. This measures the time that it takes for an interrupt card to pass through the infrared beam. Strictly speaking it measures the average speed of the card. The shorter the card is, the more accurate the measurement will be, with the limit being the thickness of the beam itself. Other methods involve fixed time intervals - of which the best technique is careful frame-by-frame analysis of video footage or GPS position data. GPS units also calculate the average speed over the short intervals between fixes from the satellite signals.
Velocity is the vector version of speed. It has a direction as well as a magnitude. The speedometer on a bike does not take into account the direction that you ride in. The vector distance (displacement) from Spanish Point to Dockyard is pretty short - about \(3\,\text{km}\). However, the distance to ride between the two is much longer - nearly \(27\,\text{km}\)! For simplicity, assume that it takes \(1.0\,\text{hr}\) to travel the distance. The average speed would be \(27\,\text{km/hr}\), whereas the velocity would be only \(3\,\text{km/hr}\).
The Kinematic Variables
For any object that is moving, especially where the motion is changing, there are 5 variables that can describe its motion. They are listed below:
For any object that is moving, especially where the motion is changing, there are 5 variables that can describe its motion. They are listed below:
Variable |
Symbol |
Units |
Comments |
Initial velocity |
\(v_o\) |
\(\text{m/s}\) |
The velocity at the start of the motion, i.e at time = zero |
Final velocity |
\(v\) |
\(\text{m/s}\) |
The velocity at the end of the motion |
Displacement |
\(x\) |
\(\text{m}\) |
How far the object has moved from its starting point - a vector |
Acceleration |
\(a\) |
\(\text{m/s}^2\) |
The rate of change of velocity |
Time |
\(t\) |
\(\text{s}\) |
How long it took - a scalar quantity |
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Acceleration is the rate of change of velocity. Students commonly think that a negative acceleration means that the object is slowing down - and often it does. But it is a bit more subtle than that. The negative sign indicates the direction of the change of velocity. It can be thought of as acting in the \(-x\) direction to the left. However, if the object is ALREADY moving to the left (i.e. has a negative velocity), then the object will move faster. If the object is moving to the right, the negative acceleration will cause it to slow down, stop and then speed up backwards.
Typically we tend to think in terms of the CARTESIAN coordinates, where up and right are positive and down and left are negative. If there is no change in direction, the signs may be ignored, but this getting into this habit could potentially lead to errors. |
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Physics is a very visual subject. There are a number of ways to visualise motion, one of the most common being the humble graph. The most useful of these is the velocity-time graph as it will give you precise information on not just the velocity but also the displacement and the acceleration of the object. They are abstract and take some getting used to. So you are encouraged to play with the simulations and games below. If you wish study calculus in the future - become a guru in these things!
Another method is a motion-diagram, where you draw a sequence of images at constant time steps to visualise the motion of the object - in a similar way to how the video freeze frame analysis works. Whether it is a graph or a diagram, try to identify which of the five variables you know, and which you need to find out.
A motion diagram of a car moving with a constant negative acceleration (i.e. decelerating)
ACTIVITIES
- There will be plenty of lab work using light gates and video analysis.
- Download a GPS app for your phone and use it to produce graphs of your journey to/from school. When you reach your destination, stop the tracking and save the data. You should be able to produce graphs of your speed and cumulative distance as a function of time. (SAFETY - Obviously DO NOT use the phone while you are riding, set the app running and then put the phone in your bag.)
- Run some of the simulations below, especially the game as they will help you understand the relationship between distance-time and velocity-time graphs.
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from Graphing Motion - interactive figures that are related to the College Physics textbook that we use. This is one of the freely available ones from Wadsworth Media.
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Graphing Motion in One-dimension. Simulation by Walter Fendt
This is rather basic but is really good to show the differences between x-t, v-t and a-t graphs. The concept of the gradient representing the rate of change should be clearer after playing with this a few times. |
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Physics Aviary Game: Use the accelerator and brake pedals to match the graphs to the motion. Good practice on gradients and time intervals.
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Couple of YouTube videos from the coolest youtube teachers in the US. Professor Dave and Mr Anderson. They have a wide range of high quality videos and if you haven't understood what I have been talking about in class they are well worth the time.
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VideoPhysics - short instructional video to the motion analysis app that we will be using a lot over the course.
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Other Resources
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Bozeman Science - Motion This is an excellent video that describes position, velocity and acceleration
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Motion Graphs - interactive graphing
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Wonders of Physics - Kinematics. Excellent cartoons from Evan Toh
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CK12 Physics - Motion - great site loaded with videos
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Tracker - a computer based version of the VideoPhysics app. A bit more flexible than the app and more accurate as you can import a video from a higher quality camera than the iPad has. The mouse pointer is easier to use! The video tutorial is a bit of a must.
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