Acceleration: Velocity Vector
Acceleration: Velocity Vector
Acceleration: Velocity Vector
Acceleration is defined as the rate of change of velocity. Acceleration is inherently a vector quantity, and an object will have non-zero acceleration if its speed and/or direction is changing. The average acceleration is given by
where the small arrows indicate the vector quantities. The operation of subtracting the initial from the final velocity must be done by vector addition since they are inherently vectors. The units for acceleration can be implied from the definition to be meters/second divided by seconds, usually written m/s2. The instantaneous acceleration at any time may be obtained by taking the limit of the average acceleration as the time interval approaches zero. This is the derivative of the velocity with respect to time:
Index
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Index A bar above any quantity indicates that it is the average value of that quantity. If the acceleration is constant, then equations 1,2 and 3 represent a complete description of the motion. Equation 4 is obtained by a combination of the others. Click on any of the equations for an example. Graphing one-dimensional motion Motion concepts
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HyperPhysics***** Mechanics
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Motion Example
Initial velocity = m/s, Final velocity = m/s
Index Distance traveled x = m In this example, the items labeled on the diagram are considered primary: if one of them is changed, the others remain the same. The data in the boxes may be changed, and the calculation will be done when you click outside the box, subject to the constraints described. Changing average velocity, acceleration or time will force a change in at least one of the original quantities. In this version, the final velocity is allowed to change. HyperPhysics***** Mechanics Distance x = Initial velocity v0 = Final velocity v = Average velocity = Acceleration a = Time t = s m m/s m/s m/s m/s^2 Motion concepts
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Motion Example
Initial velocity = m/s, Final velocity = m/s Index Motion concepts Distance traveled x = m In this example, the items labeled on m the diagram are considered primary: if Distance x = one of them is changed, the others Initial velocity v0 = remain the same. The data in the Final velocity v = boxes may be changed, and the
m/s m/s
calculation will be done when you Average velocity = click outside the box, subject to the constraints described. If the average Acceleration a = velocity is directly changed, the final Time t = s velocity is adjusted for consistency. If the acceleration or time is changed, then the distance is allowed to change.
m/s m/s^2
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m=
m/s
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Motion Graphs
Constant acceleration motion can be characterized by motion equations and Index by motion graphs. The graphs of distance, velocity and acceleration as functions of time below were calculated for one-dimensional motion using the Motion motion equations in a spreadsheet. The acceleration does change, but it is concepts constant within a given time segment so that the constant acceleration equations can be used. For variable acceleration (i.e., continuously changing), then calculus methods must be used to calculate the motion graphs.
Add annotation about the slopes of the graphs. A considerable amount of information about the motion can be obtained by examining the slope of the various graphs. The slope of the graph of position as a function of time is equal to the velocity at that time, and the slope of the graph of velocity as a function of time is equal to the acceleration. HyperPhysics***** Mechanics
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In this example where the initial position and velocity were zero, the height of the position curve is a measure of the area under the velocity curve. The height of the position curve will increase so long as the velocity is constant. As the velocity becomes negative, the position curve drops as the net positive area under the velocity curve decreases. Likewise the height of the velocity curve is a measure of the area under the acceleration curve. The fact that the final velocity is zero is an indication that the positive and negative contributions were equal. HyperPhysics***** Mechanics
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Motion Graphs
Constant acceleration motion can be characterized by motion equations and by motion graphs. The graphs of distance, velocity and acceleration as functions of time below were calculated for one-dimensional motion using the motion equations in a spreadsheet. The acceleration does change, but it is constant within a given time segment so that the constant acceleration equations can be used. For variable acceleration (i.e., continuously changing), then calculus methods must be used to calculate the motion graphs.
Add annotation about the slopes of the graphs. A considerable amount of information about the motion can be obtained by examining the slope of the various graphs. The slope of the graph of position as a function of time is equal to the velocity at that time, and the slope of the graph of velocity as a function of time is equal to the acceleration. HyperPhysics***** Mechanics
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In this example where the initial position and velocity were zero, the height of the position curve is a measure of the area under the velocity curve. The height of the position curve will increase so long as the velocity is constant. As the velocity becomes negative, the position curve drops as the net positive area under the velocity curve decreases. Likewise the height of the velocity curve is a measure of the area under the acceleration curve. The fact that the final velocity is zero is an indication that the positive and negative contributions were equal. Go Back
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On the left hand side above, the constant acceleration is integrated to obtain the velocity. For this indefinite integral, there is a constant of integration. But in this physical case, the constant of integration has a very definite meaning and can be determined as an intial condition on the movement. Note that if you set t=0, then v = v0, the initial value of the velocity. Likewise the further integration of the velocity to get an expression for the position gives a constant of integration. Checking the case where t=0 shows us that the constant of integration is the initial position x0. It is true as a general property that when you integrate a second derivative of a quantity to get an expression for the quantity, you will have to provide the values of two constants of integration. In this case their specific meanings are the initial conditions on the distance and velocity.
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