Question

An object falling in a vacuum is affected only by the gravitational force. An equation that can model a free-falling object on Earth is \(d=-4.9 t^{2},\) where \(d\) is the distance travelled, in metres, and \(t\) is the time, in seconds. An object free falling on the moon can be modelled by the equation \(d=-1.6 t^{2}\) a) Sketch the graph of each function. b) Compare each function equation to the base function \(d=t^{2}\)

Short answer

Create tables of values and graph each function. Earth's graph is steeper than Moon's. Both are downward parabolas compared to the upward parabola of the base function.

Step-by-step solution

- Identify the given equations

The first equation for free-falling object on Earth is given as \[ d = -4.9t^2 \] and the second equation for free-falling object on the Moon is given as \[ d = -1.6t^2 \]

- Create a table of values for Earth

To sketch the graph of the first equation, create a table of values by plugging different values of \( t \) into \[ d = -4.9t^2 \]. Example: | t (seconds) | d (meters) || 0 | 0 || 1 | -4.9 || 2 | -19.6 || 3 | -44.1 || 4 | -78.4 |

- Create a table of values for Moon

Now, create a similar table of values by plugging different values of \( t \) into \[ d = -1.6t^2 \]. Example: | t (seconds) | d (meters) || 0 | 0 || 1 | -1.6 || 2 | -6.4 || 3 | -14.4 || 4 | -25.6 |

- Sketch the graph for Earth

Plot the points from the table of values on a coordinate plane and sketch the curve for the equation \( d = -4.9t^2 \). The graph should be a parabola opening downwards.

- Sketch the graph for Moon

Plot the points from the table of values on a coordinate plane and sketch the curve for the equation \( d = -1.6t^2 \). The graph should also be a parabola opening downwards, but less steep compared to the Earth's graph.

- Base function comparison

The base function provided is \[ d = t^2 \]. This base function is a parabola opening upwards. Comparing this to both equations, \[ d = -4.9t^2 \] and \[ d = -1.6t^2 \], you can see that they are both vertically reflected (flipped over the x-axis) versions of the base function \[ d = t^2 \]. Additionally, the coefficient (negative) in each equation determines the width and steepness of the parabola. A larger absolute value of the coefficient (4.9 for Earth) results in a steeper parabola compared to the smaller absolute value of the coefficient (1.6 for Moon).

Key concepts

quadratic functions

A quadratic function is a type of polynomial function that takes the form of \( ax^2 + bx + c \). The general form of our free-falling object equations on Earth and the Moon are simplified to \( d = -4.9t^2 \) and \( d = -1.6t^2 \) respectively, without the linear (\( t \)) and constant terms. These equations are parabolic because their highest exponent of \( t \) is 2. This squared term is what makes the graph of a quadratic function a parabola. The negative sign in the equations indicates that these parabolas open downwards, which is typical for representing scenarios like free fall in a vacuum where the object moves downwards.

graphing parabolas

Graphing parabolas involves plotting multiple points that satisfy the quadratic equation and connecting them smoothly to reveal the parabolic shape. For the given exercises:
To graph \( d = -4.9t^2 \), we create a table of values:

\t
• \t\tWhen \( t = 0 \), \( d = 0 \)\t
\t
• \t\tWhen \( t = 1 \), \( d = -4.9 \)\t
\t
• \t\tWhen \( t = 2 \), \( d = -19.6 \)\t
\t
• \t\tWhen \( t = 3 \), \( d = -44.1 \)\t
\t
• \t\tWhen \( t = 4 \), \( d = -78.4 \)\t

The same method is used to graph \( d = -1.6t^2 \). With values:

\t
• \t\tWhen \( t = 0 \), \( d = 0 \)\t
\t
• \t\tWhen \( t = 1 \), \( d = -1.6 \)\t
\t
• \t\tWhen \( t = 2 \), \( d = -6.4 \)\t
\t
• \t\tWhen \( t = 3 \), \( d = -14.4 \)\t
\t
• \t\tWhen \( t = 4 \), \( d = -25.6 \)\t

Plot these points and connect them to sketch each parabola. Both parabolas open downwards; one is steeper (Earth) while the other is wider (Moon).

comparative analysis of functions

By comparing the equations of the free-fall motion on Earth and the Moon to the base function \( d = t^2 \), we observe some key differences:

\t
• \t\tThe base function \( d = t^2 \) forms a parabola that opens upwards.\t
\t
• \t\tThe equations \( d = -4.9t^2 \) and \( d = -1.6t^2 \) reflect these parabolas over the x-axis, creating downward-facing parabolas.\t
\t
• \t\tThe larger negative coefficient in \( d = -4.9t^2 \) results in a steeper curve compared to the smaller negative coefficient in \( d = -1.6t^2 \). This reflects how gravity on Earth is stronger, pulling objects down more rapidly compared to the Moon.\t
\t
• \t\tVertically, the larger absolute coefficient indicates a stronger gravitational pull, thus making the fall on Earth faster and resulting in a steeper graph.\t

Analyzing these differences helps to understand how variations in coefficients and signs impact the shape and orientation of parabolas in quadratic functions.