Acceleration is the first derivative of velocity relative to time. In other words, the acceleration is the same as the slope (gradient) of the velocity-time graph. Let represents the time in seconds and the speed in meters-per-second.
For :
Initial value of : at ; Hence the point on the segment: .
Slope of the velocity-time graph is the same as acceleration during that period of time: .
Find the equation of this segment in slope-point form: .
Similarly, for :
Initial value of is the same as the final value of in the previous equation at : ; Hence the point on the segment: .
Slope of the velocity-time graph is the same as acceleration during that period of time: .
Find the equation of this segment in slope-point form: .
For :
Initial value of is the same as the final value of in the previous equation at : ; Hence the point on the segment: .
Slope of the velocity-time graph is the same as acceleration during that period of time: . There's no acceleration. In other words, the velocity is constant.
Find the equation of this segment in slope-point form: .
For :
Initial value of is the same as the final value of in the previous equation at : ; Hence the point on the segment: .
Slope of the velocity-time graph is the same as acceleration during that period of time: . In other words, the velocity is decreasing.
Find the equation of this segment in slope-point form: .
For :
Initial value of is the same as the final value of in the previous equation at : ; Hence the point on the segment: .
Slope of the velocity-time graph is the same as acceleration during that period of time: . In other words, the velocity is once again constant.
Find the equation of this segment in slope-point form: .
is in the interval . Apply the equation for that interval: .