Overview of Linear Motion Graphs
Purpose: In physics, we often deal with numbers that describe how things move. These numbers can be hard to understand just by reading them in a table. That’s why we use graphs—they let us turn numbers into pictures. A graph helps us see the movement of an object more clearly. By looking at a graph, we can easily understand whether something is moving, staying still, speeding up, or slowing down. It gives us a visual story of the motion.
Types: Scientists and students use three main types of motion graphs to describe how an object moves in a straight line over time:
- Displacement-Time Graphs: These graphs show how far an object has travelled from its starting point as time goes by. The shape of the line helps us understand if the object is moving at a steady speed or changing its speed.
- Velocity-Time Graphs: These graphs show how the speed (and direction) of the object changes with time. The slope of the line shows acceleration, and the area under the graph tells us the distance covered.
- Acceleration-Time Graphs: These graphs show how the object’s acceleration changes over time. They help us understand how fast the speed of the object is changing and whether the object is speeding up or slowing down faster or slower over time.
Displacement-Time Graphs (s–t)
Axes: On a displacement-time graph, the y-axis (vertical line) shows displacement in metres (m), which means how far the object is from where it started. The x-axis (horizontal line) shows time in seconds (s), which tells us how long the object has been moving.
Gradient meaning: The slope or gradient of the line tells us the object’s velocity. If the line is slanted, the object is moving. If the slope is steep, the object is moving fast. If the slope is gentle, the object is moving slowly. This graph helps us see how fast the object moves at different times.
Gradient formula: To calculate the gradient, we take the change in displacement (how much the position has changed) and divide it by the change in time (how much time has passed). So: Gradient = change in displacement ÷ change in time. This gives us the speed, or velocity, of the object.
Straight line: A straight, slanted line means the object is moving at a constant speed. The object doesn’t speed up or slow down. It covers equal distances in equal amounts of time.
Horizontal line: If the line is flat (horizontal), it means the object is not moving at all. The displacement stays the same, which tells us the object is standing still or stationary.
Steeper gradient: When the line becomes steeper, the object is moving faster. That means it is covering more distance in the same amount of time, so its speed is increasing.
Curved line: A curved line means the object’s speed is not staying the same. The object could be speeding up (accelerating) or slowing down (decelerating). The shape of the curve tells us how the speed is changing.
Increasing gradient: If the slope of the curve gets steeper over time, it shows that the object is speeding up more and more. This means the object is accelerating.
Decreasing gradient: If the slope of the curve becomes less steep over time, the object is slowing down. This is called decelerating or negative acceleration.
Reading displacement: To find out how far the object has travelled at a particular time, look at the y-value (displacement) at that point on the graph. This tells you the object’s position relative to where it started.
Calculating displacement change: If you want to know how far the object moved between two different times, subtract the earlier displacement value from the later one. This difference tells you the distance covered during that time period.
Velocity-Time Graphs (v–t)
Axes: In a velocity-time graph, the vertical axis shows velocity in metres per second (m/s), which tells us how fast the object is moving. The horizontal axis shows time in seconds (s), which tells us how long the object has been moving.
Gradient meaning: The slope of the line tells us how the object’s velocity is changing. This is called acceleration. If the slope is going up, the object is speeding up. If it’s going down, the object is slowing down.
Gradient formula: To find the gradient, we divide the change in velocity (how much the speed changed) by the change in time (how long it took to change). So: Gradient = change in velocity ÷ change in time.
Straight line: A straight, slanted line means the object’s acceleration is steady. The object is gaining or losing speed at a constant rate.
Horizontal line: A flat line means the object is moving at the same speed the whole time. It is not getting faster or slower.
Steeper gradient: A steeper line means the object is accelerating quickly. The faster the velocity changes, the steeper the line.
Negative gradient: If the line slopes downward, it shows the object is slowing down. This is called deceleration or negative acceleration.
Area under graph: The space under the line (between the line and the x-axis) shows the distance the object has moved. This is called displacement.
Rectangular area: If the line forms a rectangle shape with the x-axis, you can calculate the displacement by multiplying velocity by time: Displacement = velocity × time.
Triangular area: If the shape under the line is a triangle, you can find the displacement using the formula for the area of a triangle: ½ × base × height.
Complex area: If the shape under the graph is made of different parts (like rectangles and triangles), break them up into smaller parts, find the area of each one, and then add them together to find the total displacement.
Acceleration-Time Graphs (a–t)
Axes: In an acceleration-time graph, the vertical axis shows acceleration in metres per second squared (m/s²), which means how fast the object’s speed is changing. The horizontal axis shows time in seconds.
Gradient meaning: The slope or gradient in this graph shows how fast the acceleration is changing. This rate of change of acceleration is called “jerk.” It tells us if the object’s acceleration is increasing or decreasing.
Horizontal line: A flat line means the object is accelerating steadily, without any change in how fast its speed is increasing.
Line at zero acceleration: If the line is exactly on the zero mark of the vertical axis, it means there is no acceleration. The object could either be staying still or moving at a constant speed.
Steeper gradient: A steep line shows that the acceleration is changing very quickly. That means the object is suddenly speeding up or slowing down.
Area under graph: The space under the line shows the change in velocity during that time. It tells us how much faster or slower the object became.
Rectangular area: If the graph makes a rectangle shape, multiply acceleration by time to find the change in velocity: Change in velocity = acceleration × time.
Positive/negative area: If the area under the graph is above the horizontal axis, it means the velocity increased. If the area is below the axis, the velocity decreased.
Interpreting Graphs
From displacement-time: The slope of the graph tells us the object’s velocity at any moment. A steep slope means fast speed; a flat line means no movement.
From velocity-time: The slope of the line tells us the object’s acceleration. A steep upward slope means the object is speeding up quickly. A flat line means the object is moving at a constant speed.
From velocity-time (area): The area under the line tells us how far the object has moved. This is useful when the velocity changes, and we still want to know the total displacement.
Motion identification: By looking at the graph’s shape and slope, we can figure out if the object is standing still, moving at a steady speed, speeding up, slowing down, or even changing its acceleration.
Using Graphs to Solve Problems
Applications: These graphs are not just for looking at—they help us solve problems too. We can use them to find important information like how far an object moved (displacement), how fast it was going (velocity), how its speed changed (acceleration), and how long the movement took (time).
Method: To solve motion problems using graphs, we:
- Use the gradient (slope) of the line to find velocity or acceleration.
- Use the area under the line to find how far the object moved (displacement).
- Read values directly from the graph for things like time and final position.