Summary of URM and UARM Characteristics
The following graphs summarise the two main motions studied in physics: uniform rectilinear motion (URM ) and uniformly accelerated rectilinear motion (UARM). Free fall has the same characteristics as UARM, while projectile motion is a combination of horizontal motion in URM and vertical motion in free fall.
| | URM | UARM |
|
Position as a function of time |
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| Velocity as a function of time |  |
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| Acceleration as a function of time |  |
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The Equation of Uniform Rectilinear Motion
To determine the velocity of an object performing a URM, the following formula must be used:
|\overrightarrow{v}=\displaystyle \frac{\triangle \overrightarrow{x}}{\triangle t}|
where
|\overrightarrow{v}| represents the object's velocity (in |\small \text {m/s}|)
|\triangle \overrightarrow{x}| represents the displacement of the object (in |\small \text {m}|)
|\triangle t| represents the variation in time (in |\small \text {s}|)
As we can see from the equation above, velocity and displacement are vectors. We can therefore deduce that the orientation of velocity will always be the same as that of displacement and vice versa.
The Equations of Uniformly Accelerated Rectilinear Motion (UARM)
| |v_{average}=\displaystyle \frac{\triangle x}{\triangle t}| | |a=\displaystyle \frac{\triangle v}{\triangle t}| |
| |v_{f}=v_{i} + a \cdot {\triangle t}| | |\triangle x= v_{i} \cdot \triangle t +\displaystyle \frac{1}{2} \cdot a \cdot {\triangle t}^{2}| |
| |\triangle x= \displaystyle \frac{(v_{i} + v_{f}) \cdot {\triangle t}}{2}| | |\triangle x= v_{f} \cdot \triangle t -\displaystyle \frac{1}{2} \cdot a \cdot {\triangle t}^{2}| |
| |{v_{f}}^2={v_{i}}^2+2 \cdot a \cdot \triangle x| |
In these formulas, the following variables are used:
| Variable | Definition | Units |
|---|---|---|
| |\triangle x = x_{f} - x_{i}| | Variation in position (distance travelled or displacement) = final position - initial position |
meters |\text {(m)}| |
| |v_{\text{average}}| | Average velocity | meters per second |\text {(m/s)}| |
| |v_{i}| | Initial velocity | meters per second |\text {(m/s)}| |
| |v_{f}| | Final velocity | meters per second |\text {(m/s)}| |
| |a| | Acceleration |
meters per square second |(\text {m/s}^2)| |
| |\triangle t = t_{f} - t_{i}| | Time variation = final time - initial time | seconds |\text {(s)}| |