Kinematics Graphical Derivation — Study Pack Flashcards

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Displacement

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Displacement is the vector measure of an object's change in position, measured along a straight line from initial to final point. Graphically, it equals the area under a velocity–time graph over the time interval.

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Velocity

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Velocity is the rate of change of displacement with respect to time and has both magnitude and direction. On a displacement–time graph, velocity is the slope; on a velocity–time graph, it is the ordinate value at each time.

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Acceleration

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Acceleration is the rate of change of velocity with respect to time. Graphically, it equals the slope of the velocity–time graph; constant acceleration appears as a straight non-horizontal line.

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Stationary

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An object is stationary if its velocity is zero; on a displacement–time graph its position is a horizontal line. On a velocity–time graph it is the horizontal line at $v=0$.

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Constant velocity

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Constant velocity means the object's speed and direction do not change; the velocity–time graph is a horizontal line. Displacement increases linearly and slope on displacement–time graph is constant.

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Constant acceleration

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Constant acceleration means velocity changes at a steady rate; the velocity–time graph is a straight line with constant slope. Displacement–time graphs under constant acceleration are parabolic.

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Velocity–time graph

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A velocity–time graph plots velocity vs. time: slope = acceleration, area under the curve = displacement. For constant acceleration the graph is linear; for constant velocity it is horizontal.

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Displacement–time graph

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A displacement–time graph shows position vs. time: slope = instantaneous velocity. Straight lines mean constant velocity; parabolas indicate constant acceleration.

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Area under curve

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The area under a velocity–time graph between two times equals the displacement over that interval. Compute area by decomposing into rectangles, triangles, or trapezoids for linear segments.

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Slope (graph)

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Graphical slope equals 'rise over run'; on velocity–time graphs it gives acceleration, on displacement–time graphs it gives velocity. Positive slope indicates increasing value; negative slope indicates decrease.

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Average velocity

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Average velocity over a time interval is total displacement divided by total time. For constant acceleration it equals $\frac{u+v}{2}$, where $u$ and $v$ are initial and final velocities.

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Equation: $v = u + at$

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This equation relates initial velocity $u$, final velocity $v$, acceleration $a$, and time $t$. It follows directly from slope of a velocity–time graph: $a = \frac{v-u}{t}$.

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Equation: $s = ut + \frac{1}{2}at^2$

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This displacement equation comes from area under a velocity–time graph split into a rectangle ($ut$) and triangle ($\frac{1}{2}at^2$). It applies for constant acceleration.

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Equation: $v^2 = u^2 + 2as$

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This equation links velocities and displacement without time. It is derived by eliminating $t$ between $v = u + at$ and $s = \frac{u+v}{2}t$.

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Equation: $s = \frac{u+v}{2}t$

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This form gives displacement as average velocity times time for constant acceleration. It follows from viewing the velocity–time area as a trapezoid.

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Guest mode

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Google Sign-in

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