 # Do particles have to move in a straight line to apply Suvat equations? on2t1inf8b 2022-07-19 Answered
Do particles have to move in a straight line to apply Suvat equations?
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Suvat equations come from the consideration that force in the system is constant. In Newtonian mechanics,
$\frac{d\mathbf{v}}{dt}=\mathbf{a}$
Where, a, is a constant. This gives the result,
$\mathbf{v}=\mathbf{a}t+\mathbf{c}$
To calculate the integration constant set t=0, and you shall have $\mathbf{c}={\mathbf{v}}_{0}$. The equation becomes,
$\mathbf{v}=\mathbf{a}t+{\mathbf{v}}_{0}$
Similarly we know that,
$\mathbf{v}=\frac{d\mathbf{r}}{dt}=\mathbf{a}t+{\mathbf{v}}_{0}$
Which gives us,
$\mathbf{r}=\frac{1}{2}\mathbf{a}{t}^{2}+{\mathbf{v}}_{0}t+\mathbf{c}$
To get the integration constant, set t=0 and the constant will be $\mathbf{c}={\mathbf{r}}_{0}$ which will give us,
$\mathbf{r}=\frac{1}{2}\mathbf{a}{t}^{2}+{\mathbf{v}}_{0}t+{\mathbf{r}}_{0}$
You can get the third equation by taking dot product of v and doing some intelligent substitution to remove time variable. Any other variation can be achieved in similar ways. Also this makes your assumption kind of right, this equation holds for particle moving in a straight line, since by moving in straight line they are exhibiting constant acceleration. However, there can also be cases when acceleration is constant, and yet motion is not in a straight line. Obviously situations where the acceleration changes direction with the object, like a car turning around, will then move outside the domain of suvat equation since there is a centripetal force working on it which is not constant in direction. However when considered the force lies outside the object, for cases like projectile motion under constant gravity condition, the object's path will not be straight, however you can apply suvat equations. Although for any sort of constant circular acceleration, that is constant angular acceleration, like the discussed car's case, or spinning pebble attached with the string, one can come up with equations that corresponds to suvat's case in constant acceleration.