Time–lapse photography is used to track the height of an object in metres. The following heights were recorded at times 0.2s, 1.2s and 2.1s respectively: 9.203m, 14.344m and 10.591m.
a) Determine an equation to model the height of the object over time, in seconds, documenting a necessary assumption.
b) Hence determine the maximum height reached by the object.
c) If the object was launched from a height of 5m and continued until it hit the ground, determine the valid domain for the function and interpret this in terms of the timing of the photography and the time the object was in the air
d) Determine the new rule for the function, including the domain, so that t=0 corresponds to the time the object was launched.
e) State any limitations of this modelling process.
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a) To determine an equation to model the height of the object over time, we can use the concept of quadratic functions. A quadratic function is in the form of y = ax^2 + bx + c, where y represents the height and x represents time. We need to assume that the object follows a parabolic path.
To find the equation, we can substitute the given data points into the quadratic function and solve for a, b, and c. Let's use the three data points to form three equations:
When t = 0.2s, h = 9.203m:
9.203 = a(0.2)^2 + b(0.2) + c
When t = 1.2s, h = 14.344m:
14.344 = a(1.2)^2 + b(1.2) + c
When t = 2.1s, h = 10.591m:
10.591 = a(2.1)^2 + b(2.1) + c
We now have a system of three equations with three variables (a, b, c). We can solve this system simultaneously using any preferred method (such as substitution or elimination) to find the values of a, b, and c, thereby giving us the equation to model the height of the object over time.
b) To determine the maximum height reached by the object, we need to find the vertex of the parabolic function. The vertex represents the highest or lowest point on the graph.
The vertex of a quadratic function in the form y = ax^2 + bx + c is given by the formula:
x = -b / (2a) and y = f(x), where f(x) is the height.
By substituting the values of a, b, and c into this formula, we can find the time (x-coordinate) and height (y-coordinate) of the vertex.
c) If the object was launched from a height of 5m and continued until it hit the ground, the valid domain for the function would be from the time of launch until the time it hits the ground. In this case, the domain would be the interval from the time the object was launched until the time it hits the ground, denoted as [t1, t2].
In terms of the timing of the photography, we would need to ensure that the time intervals we have (0.2s, 1.2s, and 2.1s) are within the valid domain [t1, t2], so we can accurately model the height over time.
The time the object was in the air would be the difference between the time it takes to launch the object and the time it takes to hit the ground.
d) To determine the new rule for the function, where t=0 corresponds to the time the object was launched, we need to shift the time axis.
Since t=0 corresponds to the time the object was launched, we must find the time difference between the launch time and the initial time recorded in the time-lapse photography. Let's say the initial time recorded in the time-lapse photography is t_initial.
The new rule for the function would then be:
h = a(t - t_initial)^2 + b(t - t_initial) + c
The domain for this function would still be the interval from the time of launch until the time the object hits the ground, [t1, t2].
e) Limitations of this modeling process could include assumptions made about the object's motion, such as assuming the path is a parabolic trajectory or neglecting factors like air resistance. Additionally, the accuracy of the model heavily depends on the accuracy of the recorded measurements and the assumption that the recorded heights correspond exactly to the object's true height at those specific times.