Graphs are used to understand motion visually. Master how to read and draw x-t graphs and v-t graphs.
Basic Knowledge
x-t graph (position-time graph):
・Horizontal axis: t(time), vertical axis: x(position or displacement)
・The slope of the graph = velocity
・Uniform linear motion → straight line, uniformly accelerated motion → parabola(quadratic curve)
・Positive slope = motion in the positive direction, negative slope = motion in the negative direction, zero slope = stationary
v-t graph (velocity-time graph):
・Horizontal axis: t(time), vertical axis: v(velocity)
・The slope of the graph = acceleration
・The area enclosed by the graph and the t-axis = displacement
・Uniformly accelerated motion → straight line, uniform linear motion → a line parallel to the t-axis
・The graph is below the t-axis = motion in the negative direction
・Passing through v=0 = turning point(change in direction)
When finding area, pay attention to the sign: above the t-axis is positive, and below it is negative displacement.
x-t graph（Relationship between position and time. Slope = velocity）
v-t graph（Relationship between velocity and time. Slope = acceleration, area = displacement）
Slope of a graph（For a straight line, (Δy)/(Δx); for a curve, the slope of the tangent）
Sign of displacement（Positive displacement = motion in the positive direction, negative displacement = motion in the negative direction）
Turning point（A point on a v-t graph where v=0 and acceleration≠0. The direction of motion reverses）
Uniform linear motion（a=0. The x-t graph is a straight line, and the v-t graph is a horizontal line）
Further Study (Background and Significance)
Why the area of a v-t graph represents displacement: The displacement during a small time interval dt is v・dt(velocity × time). Adding these quantities(integrating them) gives the “area between the graph and the t-axis,” which is the total displacement. This is the first example of the application of calculus to physics.
Why the slope of an x-t graph represents velocity: slope = Δx / Δt = displacement/time = velocity. The slope over a very small interval is the instantaneous velocity.
When reading graphs, it is important not to be “deceived by appearances.” For example, even if a curve in an x-t graph bends upward, this does not necessarily mean that the velocity is large; it means that the acceleration is positive(the velocity is increasing).
- Slope of an x-t graph = velocity
- Slope of a v-t graph = acceleration
- Area of a v-t graph = displacement
- Uniform motion → x-t is a straight line, v-t is a horizontal line
- Uniformly accelerated motion → x-t is a parabola, v-t is a straight line
- Passing through v=0 = turning point(change in direction)
- Sign of area: above the t-axis = positive displacement
Points to Note (Common Confusions)
① An curved x-t graph means that velocity changes, so acceleration exists. It is not a straight line. ② Note that the area of a v-t graph is displacement, not “distance traveled.” When there is a turn, distance traveled is found by adding the absolute values of the areas. ③ In a v-t graph, velocity can be negative while acceleration is positive(for example, while the speed is decreasing but the direction is changing). ④ The y-intercept of an x-t graph = initial position; the y-intercept of a v-t graph = initial velocity.
Practice
- In a v-t graph, the slope of the line is 3 m/s² and v=6 m/s at t=0. Find the velocity at t=4 s.
- For the v-t graph above, find the displacement from t=0 to t=4 s using area.
- Explain what kind of motion an object undergoes when its x-t graph is an upward-sloping parabola.