Particle Acceleration

Acceleration is the change in velocity over time. Refer to equation 1.

(Eq 1)  $a = \frac{dv}{dt}=\frac{d^2s}{dt^2}$

dv = Change in velocity

dt = Change in time

If there is a constant acceleration that is being applied, equations 4 and 5 could be used to determine the velocity or displacement due to the constant acceleration. Equation 2 takes in consideration of an initial velocity, and equation 3 takes in consideration of an initial velocity and displacement.

(Eq 2)  $v=v_o+a_ct$

vo = initial velocity

a c = constant acceleration

(Eq 3)  $s=s_o+v_ot+\frac{1}{2}a_ct^2$

so = initial position

For example if a particle travelling at constant acceleration of 5 m/s2 with an initial velocity of 10 m/s after 5 seconds have past what would its new velocity and position be. The initial position can be set to 0.

$v = 10 \frac{m}{s} + 5 \frac{m}{s^2} (5s) = 35 \frac{m}{s}$

$s = 0 + 10 \frac{m}{s} (5s) + \frac{1}{2} (5 \frac{m}{s^2}) (5s)^2 = 92.5 m$

The above equations only consider an acceleration in one direction. However, we live in a 3-dimensional world which means that there coudl be an acceleration in the x-direction, y-direction, and z-direction. The magnitude of this acceleration can be determined by vector algebra as seen in equation 4.

(Eq 4) $a=\sqrt{a_xi^2+a_yj^2+a_zk^2}$

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