When using the Vertical motion model, where does the leading coefficient -16 come from? When will it not be -16 and how will you know?
I'm just trying to understand the Model.
g = -32 ft/s^2 in the feet (old American - English) system so (1/2)g = -16
Watch out, most of us use metric units
g = -9.81 m/s^2
in which case
h = Hi + Vi t - 4.9 t^2
the -16 comes from the acceleration due to gravity. That is -32 ft/s^2.
The general form starting at height h with upward velocity v is
s(t) = h + vt + 1/2 at^2
where a is negative, since gravity pulls downward.
In the vertical motion model, the leading coefficient -16 comes from the acceleration due to gravity. When an object is in free fall near the surface of the Earth, its acceleration is approximately 32 feet per second squared (or 9.8 meters per second squared if using the metric system) directed downwards. However, in the vertical motion model, we consider the upward direction as positive, so we use -32 (or -9.8) as the acceleration due to gravity.
Since the vertical motion model is commonly used to represent physical situations involving objects in free fall, the leading coefficient will usually be -16 (if using feet) or -9.8 (if using meters). However, it's important to note that in some situations, the leading coefficient may be different.
For example, in situations where the object is not being influenced by gravity, such as when moving vertically in water or in outer space, the leading coefficient will not be -16 or -9.8. In such cases, the specific value for the leading coefficient would need to be provided or determined based on the given circumstances.
To know whether the leading coefficient is -16 or -9.8, you should consider the context of the problem and the physical conditions associated with the motion being modeled. If the problem involves free fall or is related to an object near the surface of the Earth, you can generally assume the leading coefficient to be -16 (or -9.8). However, when dealing with other situations, it is necessary to carefully analyze the conditions to determine the appropriate value for the leading coefficient.
The leading coefficient -16 in the vertical motion model comes from the acceleration due to gravity, which is approximately -32 feet per second squared (or -9.8 meters per second squared). This value is divided by 2, since the equation for vertical motion assumes constant acceleration over each unit of time. So, -32/2 gives us -16.
In the vertical motion model, the equation is usually written in the form of h(t) = -16t^2 + v0t + h0, where h(t) represents the height of an object at time t, v0 is the initial velocity, and h0 is the initial height.
The -16 coefficient indicates the rate at which the object's height is changing due to the force of gravity. Negative value signifies that the object is moving downwards in the model.
However, it is important to note that the value of -16 may not always be constant. If you are dealing with a different unit of measurement or different gravity than standard Earth's gravity, then the coefficient can change. For example, if you are measuring velocity in meters per second and time in seconds, the coefficient would be approximately -4.9 instead of -16.
To know the appropriate value to use, you need to consider the unit of measurement and the acceleration due to gravity specific to your scenario.