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7.2: One-Dimensional Noncontinuous Motion

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    50599
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    In continuous motion, we used a single mathematical function each to describe the position, velocity, or acceleration over time. If we cannot describe the motion with a single mathematical function over the entire time period, that motion is considered noncontinuous motion. In cases such as this, we will use different equations for different sections of the overall time period.

    For an example of noncontinuous motion, imagine a car that accelerates for a few seconds, then holds a steady speed for a few seconds, then puts on the brakes and comes to a stop over the final few seconds. There is no one mathematical function we can use to describe the motion for the full time period, but if we break the motion into three pieces, then we can come up with an equation for each section of the motion.

    Graph of a car's velocity in meters per second against time in seconds: starting from the origin it first increases at a consistent rate during Period 1, then stays at a constant value for Period 2, and finally decreases down to zero at a different consistent rate during Period 3.
    Figure \(\PageIndex{1}\): As this car accelerates, its velocity steadily goes up for a few seconds, then it holds constant for a few seconds, then it steadily goes down for a few seconds. Since we would need a separate equation for each section of this motion, this is considered noncontinuous motion.

    Analyzing the first time period will be exactly the same as analyzing a continuous function. We will initially need to identify the mathematical function to describe position, or velocity, or acceleration for that first time period. Next we take derivatives to move from position to velocity to acceleration or take integrals to move from acceleration to velocity to position. Whenever we take an integral, we need to remember to include the constant of integration which will represent the initial velocity or the initial position (in the velocity and position equations, respectively).

    For the second, third, and any other following sections, we will do much the same process. We will start by identifying an equation for the position, or velocity, or acceleration for that time period. From there we again take derivatives or integrals as appropriate, but now the constants of integration will be a little more complicated. Those constants still represent initial velocities and and positions in a sense, but they will be the velocity and position when \(t=0\), not the velocity and position at the start of that section.

    To find the constants of integration, we are instead going to have to use the transition point, which is the point in time where we are moving from one set of equations to the next. Even though the equations are changing, we cannot have an instantaneous jump in either the position or velocity. An instantaneous jump in either position or velocity would require infinite acceleration, which is physically impossible.

    Graphs showing the acceleration, velocity, and position of the car from Figure 1 above as functions of time in seconds. The initial position is depicted as 0 meters.
    Figure \(\PageIndex{1}\). Though there are noticeable jumps in the acceleration moving from one section to the next, there are no jumps in the velocity or position as we move from one section to the next. This is because jumps in those plots would require infinite accelerations.

    To find the velocity equations for the second time frame (or third, fourth, etc.), we start by integrating the acceleration equation for that same time period. This will lead to an equation with an unknown constant of integration. To solve for that constant, we look back to the velocity equation for the previous time frame and solve for the velocity at the very end of this prior time period. Since it can't jump instantaneously, this is also the velocity at the start of the current time period. Using this velocity, along with the time \(t\) at the transition point, we can solve for the last unknown in the current velocity equation (the constant of integration).

    To find the position equation for the second time frame (or third, fourth, etc.), we start by integrating the velocity equation for the same time period (you will need to solve for the unknowns in the velocity equation first, as discussed above). After integration, we should have one new constant of integration in the position equation. Just as we did with the velocity equations, we will use the position equation from the prior time frame to solve for the position at the transition point, then use that value along with the known time \(t\) to solve for the unknown constant in the current position equation.

    Video lecture covering this section, delivered by Dr. Jacob Moore. YouTube source: https://youtu.be/yNBenQRalhU.

    Example \(\PageIndex{1}\)

    A car accelerates from rest at a rate of 10 m/s2 for 10 seconds. The car then immediately begins decelerating at a rate of 4 m/s2 for another 25 seconds before coming to a stop. Find the equations for the acceleration, velocity, and position functions over the full 35-second time period, and plot these functions.

    A red racecar on a track.
    Figure \(\PageIndex{3}\): A red racecar moving along a track.
    Solution
    Video \(\PageIndex{2}\): Worked solution to example problem \(\PageIndex{1}\), provided by Dr. Majid Chatsaz. YouTube source: https://youtu.be/tPBlDQsiX_c.

    Example \(\PageIndex{2}\)

    A plane with an initial speed of 95 m/s touches down on a runway. For the first second the plane rolls without decelerating. For the next 5 seconds reverse thrust is applied, decelerating the plane at a rate of 4 m/s2. Finally, the brakes are applied with reverse thrust increasing the rate of deceleration to 8 m/s2. How long does it take for the plane to come to a complete stop? How far does the plane travel before coming to a complete stop?

    A plane moving along a runway.
    Figure \(\PageIndex{4}\): A plane is moving down an airport runway.
    Solution
    Video \(\PageIndex{3}\): Worked solution to example problem \(\PageIndex{2}\), provided by Dr. Majid Chatsaz. YouTube source: https://youtu.be/Of-ipYWblrQ.

    Example \(\PageIndex{3}\)

    A satellite's motion is described by the velocity function shown below over a sixty-second time period. For that same time period, determine the satellite's acceleration and position functions and draw these functions on a plot.

    Graph of satellite's velocity (in meters per second) vs time (in seconds) for 60 seconds. Velocity starts at the origin and is described by the function 1.5*t^2 for the first 10 seconds, and by the function 30*t-150 for the remaining 50 seconds.
    Figure \(\PageIndex{5}\): problem diagram for Example \(\PageIndex{3}\). Graph of a satellite's velocity in m/s for a period of 60 seconds, with the functions describing the velocity given.
    Solution
    Video \(\PageIndex{4}\): Worked solution to example problem \(\PageIndex{3}\), provided by Dr. Majid Chatsaz. YouTube source: https://youtu.be/ngI6_I8VFPs.

    This page titled 7.2: One-Dimensional Noncontinuous Motion is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Jacob Moore & Contributors (Mechanics Map) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.