GradeGen.AI

Acceleration and Gradients

Every time you press the accelerator in a car, you feel pushed back into your seat as your velocity changes. On a velocity-time graph, this acceleration (the rate of change of velocity) is represented by the gradient (slope) of the line.

A straight diagonal line pointing upwards shows constant (uniform) acceleration.
A straight diagonal line pointing downwards shows constant deceleration (slowing down).
A horizontal line shows a constant velocity where the object does NOT change speed, meaning acceleration is $0\text{ m/s}^2$ .
A curved line represents non-uniform acceleration.

A steeper gradient indicates a higher magnitude of acceleration because the object is changing velocity more quickly.

Worked Example: Calculating Acceleration from a Graph

A velocity-time graph shows a straight diagonal line starting at the origin $(0\text{ s}, 0\text{ m/s})$ and ending at coordinates . Calculate the acceleration.

Step 1: Extract values from the axes to find the change in velocity (y-axis) and time taken (x-axis). $\Delta v = 12\text{ m/s} - 0\text{ m/s} = 12\text{ m/s}$ $t = 4\text{ s} - 0\text{ s} = 4\text{ s}$
Step 2: Write the formula for the gradient (acceleration). $a = \frac{\Delta v}{t}$ (where $\Delta v$ is the change in velocity)
Step 3: Substitute the values. $a = \frac{12}{4}$
Step 4: Calculate the final answer with units. $a = 3\text{ m/s}^2$

For curved graphs (Higher Tier), you must find the instantaneous acceleration by drawing a tangent (a straight line that just touches the curve at a specific point). You then calculate the gradient of this tangent by forming a large right-angled triangle.

Distance from Area Under the Graph (Higher Tier)

You can figure out exactly how far a journey was just by looking at the geometric shape of a velocity-time graph. The area under the graph (the space between the velocity line and the time axis) represents the total distance travelled or displacement.

For linear graphs with straight lines, split the area into standard shapes like rectangles and triangles.
For non-linear, curved graphs, estimate the area by counting the squares under the curve.
To find the distance from squares, multiply the x-axis value of one square by the y-axis value: $\text{Value of 1 square} = (\text{width in s}) \times (\text{height in m/s})$ .

Worked Example: Calculating Distance from a Graph

A velocity-time graph shows a diagonal line rising from $(0\text{ s}, 0\text{ m/s})$ to a peak at $(5\text{ s}, 10\text{ m/s})$ , then remaining perfectly horizontal until $t = 15\text{ s}$ . Calculate the total distance travelled.

Step 1: Split the area under the line into standard geometric shapes by reading the axes. There is a triangle from $t = 0\text{ s}$ to $t = 5\text{ s}$ , and a rectangle from $t = 5\text{ s}$ to $t = 15\text{ s}$ .

Terminal Velocity

Why does a skydiver eventually stop speeding up, no matter how long they fall? This happens due to the changing balance of forces acting on an object falling through a fluid (like air or water).

Initial fall: At release, the only force is weight, acting downwards. This creates a large resultant force, causing an initial downward acceleration of approx $9.8\text{ m/s}^2$ .
Increasing speed: As the object falls faster, air resistance (drag) increases because it collides with more air particles per second.
Resultant force reduction: The weight does NOT change. However, because upward drag is increasing, the overall resultant force decreases, meaning acceleration decreases (the graph curve becomes less steep).
Terminal velocity: Eventually, air resistance increases until it exactly equals the weight. The forces are in a state of equilibrium. The resultant force becomes zero ( $0\text{ N}$ ), so the object stops accelerating and reaches a constant maximum speed, called terminal velocity. This appears as a horizontal line on the graph.

Opening a parachute drastically increases surface area, creating a sudden spike in air resistance. The drag temporarily becomes greater than weight, creating an upward resultant force that causes rapid deceleration until a new, lower terminal velocity is reached.

If you drop different stacks of cupcake cases, a heavier stack reaches a higher terminal velocity. The weight is greater, so the cases must fall faster to generate enough air resistance to balance the forces again.

Students often think an object "slows down" as air resistance initially increases during a fall, but it actually continues to speed up, just at a decreasing rate of acceleration.

In $6$ -mark questions explaining a skydiver's fall, examiners explicitly look for the phrase "resultant force becomes zero" before you conclude that terminal velocity is reached.

When calculating the gradient from a graph in the exam, draw a "large triangle" covering at least half the length of the line to reduce measurement errors and secure the method mark.

When finding distance from the area under a graph, don't forget the $\frac{1}{2}$ in the triangle formula, and always check the scale on the axes as $1$ square might represent more than $1$ unit.

Never write "constant speed" to describe a horizontal line on a velocity-time graph; AQA mark schemes strictly prefer the term "constant velocity".

The rate of change of velocity, which is represented by the gradient on a velocity-time graph and measured in m/s².

The slope of a line on a graph, calculated by dividing the change in the vertical axis by the change in the horizontal axis.

Motion where an object's velocity changes by the same amount every second, shown by a straight diagonal line on a velocity-time graph.

A negative acceleration where an object slows down at a steady, unchanging rate.

When an object moves at a steady speed in a straight line, meaning its resultant force and acceleration are both zero.

A straight line drawn to just touch a curve at a specific point, used to estimate instantaneous acceleration on a non-linear graph.

The geometric space between a plotted line and the horizontal axis, which represents distance travelled on a velocity-time graph.

The total length of the path taken by an object, regardless of direction.

The vector equivalent of distance, representing the straight-line distance and direction from a starting position to a final position.

A substance that can flow, such as a liquid or a gas (e.g., water or air).

The downward force acting on an object due to gravity (W = mg).

The single overall force acting on an object when all individual forces are combined.

A frictional force that opposes the motion of an object moving through a fluid.

The constant maximum velocity reached by a falling object when the upward force of drag exactly equals the downward force of weight.

A graph showing how the velocity of an object changes over time, where the gradient represents acceleration and the area underneath represents distance travelled.

A state in which opposing forces are perfectly balanced, resulting in a zero resultant force and zero acceleration.