To find the equation that describes the motion of the object based on the given data, we need to determine the relationship between time and distance. The data provided shows how the distance covered changes over time.
Here are the data points provided:
- At [tex]\( t = 1 \)[/tex] second, [tex]\( d = 4.0 \)[/tex] meters
- At [tex]\( t = 2 \)[/tex] seconds, [tex]\( d = 5.5 \)[/tex] meters
- At [tex]\( t = 3 \)[/tex] seconds, [tex]\( d = 7.0 \)[/tex] meters
- At [tex]\( t = 4 \)[/tex] seconds, [tex]\( d = 8.5 \)[/tex] meters
- At [tex]\( t = 5 \)[/tex] seconds, [tex]\( d = 10.0 \)[/tex] meters
We want to find a linear relationship between time ([tex]\( t \)[/tex]) and distance ([tex]\( d \)[/tex]), which can be expressed in the form:
[tex]\[ d = m \cdot t + b \][/tex]
where:
- [tex]\( m \)[/tex] represents the slope of the line (rate of change of distance with respect to time),
- [tex]\( b \)[/tex] represents the y-intercept (the distance at time [tex]\( t = 0 \)[/tex]).
By fitting a linear model to the data, the slope ([tex]\( m \)[/tex]) and the y-intercept ([tex]\( b \)[/tex]) are calculated as follows:
1. The slope ([tex]\( m \)[/tex]) is found to be [tex]\( 1.5000000000000002 \)[/tex].
2. The y-intercept ([tex]\( b \)[/tex]) is found to be [tex]\( 2.499999999999998 \)[/tex].
Therefore, the equation that describes the motion of the object is:
[tex]\[ d = 1.5 \cdot t + 2.5 \][/tex]
So, the distance covered by the object at any given time [tex]\( t \)[/tex] seconds can be calculated using the equation [tex]\( d = 1.5t + 2.5 \)[/tex].
