Week 4 Acceleartion Flashcards
Acceleration
Is the rate of change of velocity with respect to time
Mathematically, this can be expressed in the following way
A = ^v/^t=. dv/dt
The acceleration vector gives us the information on both the rate of change of speed and the rate of change of the direction of motion
Note that an object moving with a constant speed on a trajectory which is not a straight line still accelerates since the direction of motion changes.
Unit of acceleration
M/s
——
S. M/s2
Components of the acceleration vector in the Cartesian coordinate system
The components of acceleration in the Cartesian coordinate system are the rates at which the corresponding velocity components change with time. Mathematically, this can be expressed in the following way:
Ax (t) dv x
——
Dt
Components of the acceleration vector
Tangential and centripetal
Instead of resolving the acceleration vector along the axes of the Cartesian coordinate system, acceleration can be represented as the vector sum of “tangential acceleration” and “centripetal” or “normal” acceleration.
Tangential acceleration aT. accounts for the change of speed. Its direction is along the tangent to the trajectory (same as velocity), and its magnitude equals the rate of change of speed.
Centripetal acceleration aN accounts for the change in direction of motion. Its direction is towards the centre of curvature of the trajectory and it has the magnitude of the square of speed divided by the radius of curvature.
aT = dv/ dt
aN = v2/R
Changing direction
Tangential always act at at tangentes to the curve if travelling in a straight line at constant velocity it would be 0
Centripetal acting inwards toward the centre of the curve
If in straight line it would be 0
Estimating acceleration components from experimental data
Following the same logic as in the case of estimating the velocity components from position data, the components of instantaneous acceleration at some specific time “t” can be estimated from the velocity components as:
ax(t) Vx(t t) Vx (t. t)
———————
2 t (2 jumps in time)
This procedure requires that the velocity components are known
Ie that they have been calculated form the position data
With a bit of mathematics it can be shown that acceleration components can be calculated in a single step, directly from the position data using the following formulas:
Ax (t). X (t t) 2x (t). X(t t )
——————————
T 2
These formulas are actually more accurate than those shown in previous slides
Acceleration measurement systems
Acceleration can be measured “directly” using devices called accelerometers.
Direct measurement of acceleration is very useful in the context of injury assessment (e.g. crash tests). Motion reconstruction from acceleration measurement is difficult since the accelerometer axes move together with the body segments to which they are attached.
Accelerations can be used for event identification.
Acceleration can also be used to define magnitudes and directions of impacts.
Alongside number of impacts, it may provide a measure of workload.
Incremental change in velocity
The incremental change in a velocity component over a small increment of time dt is equal to the area of the rectangle with sides ax (ay or az) and dt.
Change in velocity over a time interval from acceleration data
To get the net change in a velocity component over an interval of time we have to add all the incremental changes.
This amounts to painting the area beneath the acceleration curve for that time interval.
Mathematically, this is expressed as the integral of acceleration over that period of time
Velocity from acceleration data
Velocity can be calculated from acceleration data by adding using the mathematical operation called integration.
For each velocity component this is expressed mathematically as:
Vx(t2) = vx (t1) + ax( t) dt
Previous velocity + arte of acceleration = new
velocity
In practical terms, the increment in a velocity component over a certain period of time is equal to the area beneath the curve of the corresponding acceleration component plotted against time.
Velocity from experimental acceleration data
In most cases you will be dealing with a (large) number of acceleration data points painting a curve that cannot be integrated in one go.
The solution is to break the total area into rectangles or trapezoids whose heights are equal to measured accelerations and whose widths are equal to the measurement time interval (Dt). The net area is the sum of areas of all the rectangles or trapezoids.
Key points
Acceleration is the rate of change of velocity with time
Acceleration can be resolved into components along the axes, or into the tangential and centripetal component. The tangential component is due to the change of speed and the centripetal component is due to change in direction.
Acceleration can be calculated from experimental position data
Velocity and position can be estimated from the experimental acceleration data
