⚾️Honors Physics Unit 2 Review

Velocity-time graphs give you a visual way to extract almost everything about an object's motion: its displacement, acceleration, and how its speed changes over time. The two key skills are reading the slope (which gives acceleration) and finding the area under the curve (which gives displacement). Once you're comfortable with those, you can move fluently between position-time and velocity-time representations.

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Velocity-Time Graph Interpretation

Slope = Acceleration

The slope at any point on a v-t graph tells you the object's acceleration at that moment.

A positive slope means velocity is increasing (positive acceleration).
A negative slope means velocity is decreasing (negative acceleration).
A zero slope (horizontal line) means constant velocity, so acceleration is zero.
A steeper slope means a greater magnitude of acceleration.

One common mistake: a negative slope doesn't always mean the object is slowing down. If the object has negative velocity and the slope is also negative, the object is actually speeding up in the negative direction. What matters is whether the velocity and acceleration share the same sign.

Area Under the Curve = Displacement

Area above the time axis counts as positive displacement (motion in the positive direction).
Area below the time axis counts as negative displacement (motion in the negative direction).
Net displacement is the algebraic sum of positive and negative areas.
Total distance traveled is the sum of the absolute values of those areas. This is different from net displacement whenever the object reverses direction.

Instantaneous velocity is read directly off the graph at any point in time.

Velocity-time graph interpretation, Acceleration | Boundless Physics

Calculations from Velocity-Time Graphs

Average Velocity

$v_{avg} = \frac{\Delta x}{\Delta t}$

where $\Delta x$ is the net displacement (area under the curve, with sign) and $\Delta t$ is the total time interval.

Finding Net Displacement Step-by-Step

Break the area under the curve into simple geometric shapes: rectangles, triangles, and trapezoids.
Calculate the area of each shape. For a triangle: $A = \frac{1}{2}bh$ . For a trapezoid: $A = \frac{1}{2}(b_1 + b_2)h$ .
Assign each area a sign: positive if above the time axis, negative if below.
Add all the signed areas together. The result is the net displacement.

For example, if a v-t graph shows a triangle above the axis with area +12 m and a triangle below the axis with area −4 m, the net displacement is +8 m, but the total distance traveled is 16 m.

Velocity-time graph interpretation, Average and Instantaneous Acceleration – University Physics Volume 1

Converting Between Position and Velocity Graphs

These two graph types are closely linked, and you should be able to go in either direction.

Position-Time → Velocity-Time

The slope of a position-time graph at any point gives the velocity at that moment.

Positive slope → positive velocity
Negative slope → negative velocity
Steeper slope → greater speed
A curved position-time graph means the velocity is changing, so you'd find the slope of the tangent line at each point to build the v-t graph.

Velocity-Time → Position-Time

The area under a velocity-time graph up to a given time gives the displacement from the starting position.

Accumulate the signed area from $t = 0$ to each point in time.
Add that displacement to the initial position to get the position at each moment.
A constant velocity (horizontal v-t line) produces a straight, sloped position-time graph.
A linearly changing velocity (sloped v-t line) produces a curved (parabolic) position-time graph.

Kinematics and Motion Analysis

Kinematics is the branch of physics that describes motion without worrying about the forces that cause it. You focus purely on quantities like position, velocity, and acceleration.

Keep the distinction between vector and scalar quantities clear:

Velocity is a vector (has magnitude and direction). It can be positive or negative on a v-t graph.
Speed is a scalar (magnitude only). It's always the absolute value of velocity.

Velocity-time graphs tie together all the major kinematic ideas. The graph itself shows velocity, its slope gives acceleration, and its area gives displacement. That makes v-t graphs one of the most information-dense tools you'll use in this course.

⚾️Honors Physics Unit 2 Review