The image displays a set of graphs illustrating the relationships between position ($x$), velocity ($v$), and acceleration ($a$) for objects moving with
constant acceleration. The graphs are organized into four vertical columns, each representing a different motion scenario.
Here is the step-by-step analysis of the patterns shown in each column:
General Rules for Constant Acceleration
1.
Acceleration ($a$) vs. Time ($t$): Since acceleration is constant, this graph is always a
horizontal line. If the line is above the axis, acceleration is positive. If below, it is negative.
2.
Velocity ($v$) vs. Time ($t$): Velocity changes at a constant rate. This graph is always a
straight diagonal line. The slope of this line equals the acceleration.
3.
Position ($x$) vs. Time ($t$): Position changes based on velocity. This graph is always a
curve (parabola).
* If acceleration is positive, the curve opens upward (like a cup $\cup$).
* If acceleration is negative, the curve opens downward (like a frown $\cap$).
---
Column 1 Analysis (Far Left)
*
Acceleration ($a$): The green line is horizontal and
below the axis. This means
negative constant acceleration.
*
Velocity ($v$): The blue line is straight and slopes
downward. It starts with a positive velocity and decreases toward zero. This matches negative acceleration (slowing down while moving forward).
*
Position ($x$): The purple curve is part of a "frown" shape (concave down). The slope starts steep (fast) and gets flatter (slowing down). This matches an object slowing down.
Column 2 Analysis (Second from Left)
*
Acceleration ($a$): The green line is horizontal and
below the axis. This means
negative constant acceleration.
*
Velocity ($v$): The blue line is straight and slopes
downward, but it is entirely
below the axis. This means the object has negative velocity and is getting more negative (speeding up in the negative direction).
*
Position ($x$): The purple curve is part of a "frown" shape (concave down). The slope starts flat (slow) and gets steeper downward (speeding up in the negative direction).
Column 3 Analysis (Third from Left)
*
Acceleration ($a$): The green line is horizontal and
above the axis. This means
positive constant acceleration.
*
Velocity ($v$): The blue line is straight and slopes
upward. It starts with a negative velocity and moves toward zero. This matches positive acceleration (slowing down while moving backward).
*
Position ($x$): The purple curve is part of a "cup" shape (concave up). The slope starts steep downward (fast backward) and flattens out (slowing down).
Column 4 Analysis (Far Right)
*
Acceleration ($a$): The green line is horizontal and
above the axis. This means
positive constant acceleration.
*
Velocity ($v$): The blue line is straight and slopes
upward, starting from zero or a positive value. This means the object is speeding up in the positive direction.
*
Position ($x$): The purple curve is part of a "cup" shape (concave up). The slope starts flat (slow) and gets steeper upward (speeding up).
Final Conclusion
The image correctly demonstrates that for constant acceleration:
* The
$a-t$ graph is a horizontal line.
* The
$v-t$ graph is a straight line with a slope equal to the acceleration.
* The
$x-t$ graph is a parabola, curving up for positive acceleration and down for negative acceleration.
Parent Tip: Review the logic above to help your child master the concept of graphing acceleration worksheet.
