Air resistance and drag can either be proportional to the velocity or to the square of the velocity. Drag force will eventually counteract downward forces on an object in freefall, resulting in a terminal velocity. The acceleration of the object can be modeled by an exponential decay graph.
PURPOSE
Evaluate how terminal velocity varies with mass.
Determine in which instances air resistance is proportional to velocity or to velocity squared.
Understand how mass influences the decay constant k.
PROCEDURE (EXPERIMENT #1)
Begin with five coffee filters layered on top of each other, holding the bunch a significant height above a motion sensor.
Drop the filters, ensuring contact right above the motion sensor. A section of the data points should
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Equip the cart with a powerful magnet that is as close to the ramp as possible without actually touching it. Have a motion sensor at the top of the ramp.
Let the cart roll down the ramp, and again obtain the terminal velocities by finding the linear section of data points.
Add a block of mass on top of the cart and repeat the experiment. Continue until you have a total of 3 blocks on top of the cart.
The three previous graphs illustrate Terminal Velocity vs Mass, Terminal Velocity Squared vs Mass, and Log Terminal Velocity vs Log Mass. An original set of data was used and then manipulated to obtain velocity squared and log terminal velocity. From the graphs and data sets it is apparent that mass is proportional to the square of the terminal velocity. Thus, for objects in freefall, drag varies with velocity squared.
Unlike the coffee filters, the mass of the car is proportional to terminal velocity. In this instance, drag varies directly with the velocity.
As we can see, additional mass will produce longer half-lives. The graph above clearly illustrates how, as mass increases, so does the average half-life. Understanding the equation of terminal velocity, whose general form is V(t) = 1-e-k*t, it is apparent that smaller k values equals longer half-lives. Therefore, the greater the mass, the smaller the k
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The first graph of experiment #2 (Terminal Velocity vs Mass) has a linear velocity slope, indicating a constant acceleration regardless of the mass. Additionally, it also means that half-life time follows the formula: t(½) = ln(2) / k. Therefore, if one were to double the half-life time, they would have to divide the k value by two, or multiply the mass by two since the mass varies inversely with the k constant (as discussed above). Hence, doubling the mass would therefore also double the half-life. Since the two are directly related, a linear trendline was placed to demonstrate that
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