Kinematics: Describing Motion

The Problem of Describing Motion

Before Newton could explain why objects move, scientists needed a precise language to describe how they move. Kinematics is that language — a purely mathematical description of motion that doesn't ask about causes.

This distinction matters. A ball thrown upward and a ball dropped from rest both follow the same kinematic rules. Kinematics doesn't care why they're moving; it just describes the motion with mathematical precision.

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Position, Displacement, and Distance

Position tells you where an object is relative to a chosen reference point (the origin). It's a vector quantity — it has both magnitude and direction.

Displacement is the change in position:

$$\Delta x=x_f-x_i$$

Notice: displacement depends only on start and end points, not the path taken. If you walk 3 m east and 3 m west, your displacement is zero — but your distance traveled is 6 m.

This is the first of many vector/scalar distinctions in AP Physics 1. Distance is scalar (always positive); displacement is a vector (can be negative, indicating direction).

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Velocity and Speed

Average velocity is displacement divided by time:

$$\bar{v}=\frac{\Delta x}{\Delta t}$$

Average speed is total distance divided by time — always positive, no direction.

Instantaneous velocity is the velocity at a single moment. On a position-time graph, instantaneous velocity equals the slope of the tangent line at that point. Average velocity equals the slope of the secant line connecting two points.

This is where calculus appears: instantaneous velocity is formally defined as:

$$v=\underset{\Delta t\to 0}{\lim }\frac{\Delta x}{\Delta t}=\frac{dx}{dt}$$

AP Physics 1 doesn't require calculus notation, but understanding this limit concept deepens your interpretation of motion graphs.

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Acceleration

Acceleration is the rate of change of velocity:

$$a=\frac{\Delta v}{\Delta t}$$

Acceleration is a vector. A car slowing down while moving forward has negative acceleration (deceleration). A car turning left at constant speed is also accelerating — because velocity's direction is changing, even though its magnitude isn't.

On a velocity-time graph:

• Slope = acceleration
• Area under the curve = displacement

This area interpretation is powerful. Even without integration, you can calculate displacement from a v-t graph by finding geometric areas (triangles, rectangles, trapezoids).

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Concept Check

A velocity-time graph shows a straight line from v = 0 at t = 0 to v = 20 m/s at t = 4 s. What is the acceleration, and what is the displacement during this interval?

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Acceleration = slope = (20 - 0)/(4 - 0) = 5 m/s². Displacement = area under the triangle = ½ × base × height = ½ × 4 s × 20 m/s = 40 m.

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The Kinematic Equations

For constant acceleration only, four equations relate the five kinematic variables: position x, initial velocity v₀, final velocity v, acceleration a, and time t.

$$v=v_0+at$$

$$x=x_0+v_{0}t+\frac{1}{2}a{t}^2$$

$$v^2=v_0^2+2a\Delta x$$

$$\bar{v}=\frac{v+v_0}{2}$$

Strategy: Identify the three known quantities and the one unknown. Choose the equation that contains exactly those four variables. The fifth variable (the one not in your chosen equation) is your "don't care" variable.

Free fall is a special case: near Earth's surface, gravity provides constant downward acceleration of g = 9.8 m/s² (use 10 m/s² for quick estimates). Choose downward as positive or negative — just be consistent.

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Reading Motion Graphs

AP exam free-response questions frequently ask you to construct or interpret graphs. Develop these automatic pattern recognitions:

Position-time graphs:

• Flat line → object at rest
• Constant slope → constant velocity
• Curved line → changing velocity (acceleration)
• Slope of tangent = instantaneous velocity

Velocity-time graphs:

• Flat line → constant velocity (zero acceleration)
• Constant slope → constant acceleration
• Area = displacement (above x-axis = positive displacement; below = negative)
• Crossing x-axis = object changes direction

Acceleration-time graphs:

• Flat line → constant acceleration (most common in AP Physics 1)
• Area = change in velocity

Interactive geometry — loading...

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Experimental Design in Kinematics

AP Physics 1 assessments often ask you to design experiments. For kinematics:

Common setups: Motion detectors (ultrasonic), video analysis (frame-by-frame), ticker tape timers, photogate timers

Key experimental skills:

• Identify controlled, independent, and dependent variables
• Recognize sources of error (friction, air resistance, timing precision)
• Propose how to linearize non-linear data (e.g., plot x vs. t² to get a straight line for constant acceleration)

When acceleration is constant, x vs. t² gives a straight line with slope a/2. This linearization technique appears repeatedly in AP lab questions.

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Concept Check

A student drops a ball from different heights and measures the time to fall. She plots height (h) on the y-axis and time squared (t²) on the x-axis. What does the slope of the best-fit line represent physically?

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For free fall, h = ½gt², so h vs. t² gives slope = g/2 ≈ 4.9 m/s². The slope is half the gravitational acceleration.

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Unit Summary

Kinematics provides the mathematical scaffolding for all of mechanics. The key ideas:

• Vectors vs. scalars: Displacement and velocity have direction; distance and speed do not
• The graph is the physics: Slope of x-t graph = velocity; slope of v-t graph = acceleration; area under v-t graph = displacement
• Kinematic equations: Valid only for constant acceleration; identify knowns and choose the right equation
• Free fall: Constant downward acceleration g ≈ 9.8 m/s²; symmetry between upward and downward phases
• Experimental literacy: Linearizing data, identifying variables, estimating uncertainty

In the next unit, we extend kinematics into two dimensions using vectors — the essential tool for analyzing projectile motion.