Objectives

• Understand the graphical relationship between position, velocity and acceleration
• Extract velocity and acceleration of an object from graphs
• Learn how to apply the constant acceleration model to a complicated motion

Resources

• Cart with fan attachment
• Flat track
• Ruler
• Stopwatch, wristwatch or stopwatch app
• Motion sensor
• Science Workshop interface and DataStudio software

Background

In a situation involving motion with constant acceleration, the position of an object is described by x=x_0+v_0t+\tfrac{1}{2}at^2

If the initial velocity and position are both zero, this equation can easily be solved for acceleration to obtain a=\frac{2x}{t^2}

For a single measurement, uncertainty must be estimated based on the accuracy of the instrument, skill of the experimenter, environmental factors, etc. If multiple repeated measurements are made, the standard deviation of the mean can provide a reasonable estimate of the uncertainty.

\sigma=\sqrt{\frac{1}{N-1} \left[ (x_1-\mu)^2+(x_2-\mu)^2+\cdots+(x_N-\mu)^2 \right] } \mathrm{SDOM}=\sigma/\sqrt{N}

To obtain the uncertainty in a quantity calculated from measurements, the uncertainty in the measurements must be propagated. To propagate the uncertainty in x and t in the equation above, this is

\left(\frac{\sigma_a}{a}\right)^2=\left(\frac{\sigma_x}{x}\right)^2+2\left(\frac{\sigma_t}{t}\right)^2

See Measurement Uncertainty for details.