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8.3: Acceleration

Last updated

Aug 8, 2024
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- 8.2: Velocity
- 8.4: Higher Derivatives

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In a similar way, we can take the derivative velocity with respect to time to get acceleration, which is the second derivative of $x$ with respect to $t$ :

$a=\frac{d v}{d t}=\frac{d^{2} x}{d t^{2}}$

SI units of acceleration are meters per second squared $\left(\mathrm{m} / \mathrm{s}^{2}\right)$ .

Example. In the previous example, we found a formula for the velocity of a particle as $v(t)=10 t$ . The acceleration of the particle in this example is $a(t)=10 \mathrm{~m} / \mathrm{s}^{2}$ , a constant.

As we'll see later when we discuss gravity, all objects at the surface of the Earth will accelerate downward with the same acceleration, $9.80 \mathrm{~m} / \mathrm{s}^{2}$ . This important constant is called the acceleration due to gravity, and is given the symbol $g$ :

$g=9.80 \mathrm{~m} / \mathrm{s}^{2} \quad\left(=32 \mathrm{ft} / \mathrm{s}^{2}\right) .$

This value is an average for the Earth; for a more exact value of $g$ , you can use Helmert's equation (Section 51.4).

The acceleration due to gravity gives rise to a common (non-SI) unit of acceleration, also called the $g$ :

$1 \mathrm{~g}=9.80665 \mathrm{~m} / \mathrm{s}^{2} .$

This number is a standardized conventional value that has been adopted by international agreement.

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