Variable-Acceleration Motion

01Variable-Acceleration Motion

Functions of Time

Describe displacement, velocity and acceleration using functions with a stated time domain.

A formula is a model over an interval

In one-dimensional motion, position $x(t)$ , velocity $v(t)$ and acceleration $a(t)$ may vary with time. State the time units, the coordinate’s positive direction and the interval on which the model applies.

v(t)=\frac{dx}{dt},\qquad a(t)=\frac{dv}{dt}=\frac{d^2x}{dt^2}

If $x(0)$ is not zero, displacement from the starting point is $x(t)-x(0)$ . Speed is $|v(t)|$ . SUVAT applies only when the acceleration function is constant over the interval.

02Variable-Acceleration Motion

Using Differentiation

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Differentiate once for velocity, twice for acceleration

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03Variable-Acceleration Motion

Using Integration

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Each integration needs its own constant

v(t_2)-v(t_1)=\int_{t_1}^{t_2}a(t)\,dt

x(t_2)-x(t_1)=\int_{t_1}^{t_2}v(t)\,dt

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04Variable-Acceleration Motion

Rest & Direction Changes

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Test the sign on either side of rest

v=t^2-6t+8

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\mathrm{m\,s^{-1}}

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Time interval	Velocity and direction
$0\le t<2$	Bqbxpapt
$2<t<4$	Ckjdhfxc
$4<t\le5$	Nbwmnnwn

05Variable-Acceleration Motion

Displacement & Distance

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Integrate over intervals of one velocity sign

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06Variable-Acceleration Motion

Maximum & Minimum Values

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Differentiate the quantity being optimised

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