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Equations of Motion

We have seen that, given the position function for an object in motion, s (t), we can find the velocity function, v(t), by taking the derivative of s and can find the acceleration function by taking the derivative of v.

Instead, suppose we are given the acceleration function and the initial velocity and position. Using antiderivatives we can then find v and s. Let a(t) be the acceleration function, v0 be the initial velocity and s0 be the initial position. Solving the differential equation dv/dt = a(t) will give us v(t) (where the constant of integration, C, is v0 in this case). Once we have v, we can then solve ds/dt = v(t) to find the position function (where the constant of integration, C, is s0 in this case).

See About the calculus applets for operating instructions.
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  • The applet shows the graph of a(t) on the left, v(t) in the middle, and s(t) on the right. In this example, the acceleration is constant and equal to -9.8 m/sec² (i.e., the downward acceleration due to gravity at the surface of the Earth). The area "under" this straight horizontal line is shown in red on the left graph from 0 to time t. You can move the slider to change the time. Note that the area on the left graph is the same as the value of the velocity in the middle graph; velocity is the antiderivative (or accumulation) of acceleration. In this example: a(t) = dv/dt = -9.8 so finding the antiderivative we get v(t) = -9.8t+v0 This is the straight line graphed in the center.

  • Now look at the area shaded in the middle graph and notice that it is the same as the value of the position function shown on the right; position is the antiderivative (or accumulation) of velocity. In this example, v(t)=ds/dt=-9.8t+v0 so finding the antiderivative gives us s(t)=-9.8/2*t^2+v0*t+s0 which is the parabola graphed on the right. It is worth remembering that this parabola is not the path of the object in motion. We are looking at one-dimensional motion here and the right hand graph shows position (e.g., height above the ground) versus time.

  • Move the v0 slider to see the effect of changing the starting velocity (i.e., v at t = 0). Move the s0 slider to see the effect of changing the starting position (i.e., s at t = 0).

    Explore

    You can enter your own acceleration function to explore more complex examples. Each graph can be panned and zoomed independently.

    Other 'Constructing Antiderivatives' topics

    Acknowledgements

    Derived from the work of Thomas S. Downey under a Creative Commons Attribution 3.0 License.

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