Speed, Velocity and Motion Graphs for SS 1

Speed, Velocity, and Motion Graphs

Performance Objectives

By the end of the lesson, students should be able to:

1. distinguish between speed and velocity;
2. state the SI units of speed and velocity;
3. measure and calculate speed and velocity of a moving body;
4. plot distance-time, and displacement-time graphs and deduce speed and velocity from the gradient of the graphs.

Understanding Motion

Motion is a fundamental concept in physics that describes a change in position of an object over time. To understand motion fully, it is important to distinguish between how far an object travels and its change in position relative to a starting point.

Distance

Distance is the total length of the path covered by a moving object, regardless of the direction of motion.

It is a scalar quantity, meaning it only has magnitude (size) and no direction. For example, if a person walks 5 metres forward and then 3 metres backward, the total distance covered is 8 metres.

The SI unit of distance is the metre (m).

Displacement

Displacement is the change in position of an object in a specific direction from its starting point to its final point.

It is a vector quantity, meaning it has both magnitude and direction. For example, if a person walks 5 metres east and then 3 metres west, their displacement is 2 metres east from their starting point.

The SI unit of displacement is the metre (m).

Speed

Speed is the rate at which an object covers distance.

It is a scalar quantity, meaning it only describes how fast an object is moving, without indicating the direction. For example, a car moving at 60 kilometres per hour (km/h) has a speed of 60 km/h.

The formula for speed is:

Speed = Distance / Time

The SI unit of speed is metres per second (m/s).

Velocity

Velocity is the rate at which an object changes its displacement.

It is a vector quantity, meaning it describes both how fast an object is moving and in what direction. For example, a car moving at 60 kilometres per hour (km/h) to the east has a velocity of 60 km/h east.

The formula for velocity is:

Velocity = Displacement / Time

The SI unit of velocity is metres per second (m/s).

Distinguishing Speed and Velocity

The main difference between speed and velocity lies in their nature as scalar or vector quantities.

Speed only tells how fast an object is moving, while velocity tells both how fast and in what direction. An object can have a constant speed but a changing velocity if its direction of motion changes, such as a car moving around a circular track at a steady rate.

The SI unit for both speed and velocity is metres per second (m/s).

Calculating Speed and Velocity

Calculations of speed and velocity involve applying their respective formulas using the given distance, displacement, and time values.

Speed Calculations

To calculate speed, divide the total distance covered by the time taken.

Example 1: A student walks a distance of 400 metres in 80 seconds. Calculate the student’s speed.

Solution:

1. Given: Distance = 400 m, Time = 80 s
2. Formula: Speed = Distance / Time
3. Calculation: Speed = 400 m / 80 s = 5 m/s
4. The student’s speed is 5 m/s.

Velocity Calculations

To calculate velocity, divide the displacement by the time taken. Remember to include the direction in the final answer.

Example 2: A cyclist travels 200 metres east in 25 seconds. Calculate the cyclist’s velocity.

Solution:

1. Given: Displacement = 200 m east, Time = 25 s
2. Formula: Velocity = Displacement / Time
3. Calculation: Velocity = 200 m east / 25 s = 8 m/s east
4. The cyclist’s velocity is 8 m/s east.

Example 3: A car travels 150 km north in 2 hours, then turns around and travels 50 km south in 1 hour. Calculate the average speed and average velocity for the entire journey.

Solution:

1. For Average Speed:
2. Total Distance = 150 km + 50 km = 200 km
3. Total Time = 2 hours + 1 hour = 3 hours
4. Average Speed = Total Distance / Total Time = 200 km / 3 hours = 66.67 km/h
5. For Average Velocity:
6. Displacement = 150 km north – 50 km south = 100 km north (from the starting point)
7. Total Time = 3 hours
8. Average Velocity = Displacement / Total Time = 100 km north / 3 hours = 33.33 km/h north

Practical Determination of Speed and Velocity

Speed and velocity can be measured in a laboratory or real-world setting using simple equipment.

Measuring Speed with a Stopwatch and Meter Rule

This procedure allows for the measurement and calculation of the average speed of a moving object like a toy car or trolley.

The required materials are:

1. Trolley or toy car
2. Stopwatch
3. Meter rule
4. Masking tape or chalk

The procedure involves the following steps:

1. Mark a starting line (A) and a finishing line (B) on a flat surface, such as a laboratory bench or classroom floor, using masking tape or chalk.
2. Measure the distance between line A and line B accurately using the meter rule. Record this distance.
3. Place the trolley or toy car at the starting line (A).
4. Start the stopwatch simultaneously as the trolley begins to move from line A.
5. Stop the stopwatch exactly when the front of the trolley crosses the finishing line (B). Record the time taken.
6. Repeat the experiment at least three times to obtain multiple readings for time. This helps to reduce errors.
7. Calculate the average time from the recorded readings.
8. Use the formula Speed = Distance / Average Time to calculate the average speed of the trolley.
9. Ensure the surface is level to avoid unintended acceleration due to gravity.

Using a Ticker Timer to Determine Speed

A ticker timer is a device that makes dots on a paper tape at regular, short intervals, usually 0.02 seconds per dot. This allows for precise measurement of time and distance.

The required materials are:

1. Ticker timer
2. Power supply (e.g., 12 V AC)
3. Paper tape
4. Trolley or toy car
5. Meter rule

The procedure involves the following steps:

1. Set up the ticker timer and thread a length of paper tape through it.
2. Attach one end of the paper tape to the back of the trolley or toy car.
3. Switch on the ticker timer.
4. Push the trolley so that it moves across a flat surface, pulling the paper tape through the ticker timer.
5. After the trolley has moved, switch off the ticker timer and detach the tape.
6. Examine the tape. It will have a series of dots. Each dot represents a fixed time interval (e.g., 0.02 s).
7. Count the number of spaces between the dots. For example, 50 spaces represent 50 x 0.02 s = 1 second.
8. Measure the length of the tape corresponding to a certain number of spaces (e.g., 50 spaces for 1 second) using the meter rule. This length is the distance covered in that time.
9. Calculate the average speed using the formula Speed = Distance / Time.
10. For more detailed analysis, divide the tape into sections (e.g., 10-dot intervals) and calculate the speed for each section to observe changes in speed.

Motion Graphs

Graphs are powerful tools in physics for visualising and analysing motion. They provide a clear representation of how distance or displacement changes over time.

Introduction to Motion Graphs

In motion graphs, time is typically plotted on the horizontal axis (x-axis), and distance or displacement is plotted on the vertical axis (y-axis).

The shape and slope of the line on these graphs reveal important information about the object’s motion, such as whether it is at rest, moving at a constant speed, or changing its speed or velocity.

Distance-Time Graphs

A distance-time graph shows how the total distance covered by an object changes over time.

The gradient (slope) of a distance-time graph represents the speed of the object. A steeper gradient indicates a higher speed, while a less steep gradient indicates a lower speed.

Interpreting Distance-Time Graphs:

1. Object at Rest: A horizontal line on a distance-time graph indicates that the object’s distance from the starting point is not changing. This means the object is stationary or at rest, and its speed is zero.
2. Constant Speed: A straight line with a positive slope indicates that the object is moving at a constant speed. The distance covered increases uniformly with time.
3. Increasing Speed (Acceleration): A curve bending upwards (getting steeper) indicates that the object’s speed is increasing. The distance covered increases at an increasing rate.
4. Decreasing Speed (Deceleration): A curve flattening out (getting less steep) indicates that the object’s speed is decreasing. The distance covered increases at a decreasing rate.

To determine speed from the gradient of a distance-time graph, select two points on the straight line part of the graph. Calculate the change in distance (rise) and divide it by the corresponding change in time (run).

Gradient (Speed) = Change in Distance / Change in Time = (d₂ – d₁) / (t₂ – t₁)

Worked Example: Plotting a Distance-Time Graph and Determining Speed

A car travels according to the following data:

Plot a distance-time graph for the car’s motion and determine its speed.

Solution:

1. Plotting the Graph:
   1. Draw two perpendicular axes. Label the horizontal axis “Time (s)” and the vertical axis “Distance (m)”.
   2. Choose appropriate scales for both axes. For time, 1 cm could represent 1 s. For distance, 1 cm could represent 5 m.
   3. Plot the given points (0,0), (2,10), (4,20), (6,30), (8,40) on the graph paper.
   4. Draw a straight line connecting these points.
2. Determining Speed from Gradient:
   1. Select two points on the straight line, for example, Point 1 (t₁=2 s, d₁=10 m) and Point 2 (t₂=8 s, d₂=40 m).
   2. Calculate the change in distance (rise): Δd = d₂ – d₁ = 40 m – 10 m = 30 m.
   3. Calculate the change in time (run): Δt = t₂ – t₁ = 8 s – 2 s = 6 s.
   4. Calculate the speed (gradient): Speed = Δd / Δt = 30 m / 6 s = 5 m/s.
3. The speed of the car is 5 m/s.

Displacement-Time Graphs

A displacement-time graph shows how the displacement of an object from its starting point changes over time, including its direction.

The gradient (slope) of a displacement-time graph represents the velocity of the object. A positive gradient indicates positive velocity (moving in the positive direction), while a negative gradient indicates negative velocity (moving in the negative direction or opposite direction). A steeper gradient indicates a higher magnitude of velocity.

Interpreting Displacement-Time Graphs:

1. Object at Rest: A horizontal line indicates that the object’s displacement is not changing. The object is at rest, and its velocity is zero.
2. Constant Positive Velocity: A straight line with a positive slope indicates constant velocity in the positive direction.
3. Constant Negative Velocity: A straight line with a negative slope indicates constant velocity in the negative direction (moving back towards or past the starting point).
4. Increasing Velocity: A curve bending upwards (getting steeper) indicates that the velocity is increasing in the positive direction.
5. Decreasing Velocity: A curve flattening out (getting less steep) indicates that the velocity is decreasing in the positive direction.

To determine velocity from the gradient of a displacement-time graph, select two points on the straight line part of the graph. Calculate the change in displacement (rise) and divide it by the corresponding change in time (run).

Gradient (Velocity) = Change in Displacement / Change in Time = (s₂ – s₁) / (t₂ – t₁)

Worked Example: Plotting a Displacement-Time Graph and Determining Velocity

An object moves according to the following data:

Plot a displacement-time graph for the object’s motion and determine its velocity during the first 6 seconds and from 6 to 12 seconds.

Solution:

1. Plotting the Graph:
   1. Draw two perpendicular axes. Label the horizontal axis “Time (s)” and the vertical axis “Displacement (m)”.
   2. Choose appropriate scales for both axes. For time, 1 cm could represent 1 s. For displacement, 1 cm could represent 5 m.
   3. Plot the given points (0,0), (3,15), (6,30), (9,15), (12,0) on the graph paper.
   4. Draw straight lines connecting these points.
2. Determining Velocity from Gradient (0 to 6 seconds):
   1. Select two points on the first straight line segment, for example, Point 1 (t₁=0 s, s₁=0 m) and Point 2 (t₂=6 s, s₂=30 m).
   2. Calculate the change in displacement: Δs = s₂ – s₁ = 30 m – 0 m = 30 m.
   3. Calculate the change in time: Δt = t₂ – t₁ = 6 s – 0 s = 6 s.
   4. Calculate the velocity (gradient): Velocity = Δs / Δt = 30 m / 6 s = 5 m/s.
3. Determining Velocity from Gradient (6 to 12 seconds):
   1. Select two points on the second straight line segment, for example, Point 1 (t₁=6 s, s₁=30 m) and Point 2 (t₂=12 s, s₂=0 m).
   2. Calculate the change in displacement: Δs = s₂ – s₁ = 0 m – 30 m = -30 m.
   3. Calculate the change in time: Δt = t₂ – t₁ = 12 s – 6 s = 6 s.
   4. Calculate the velocity (gradient): Velocity = Δs / Δt = -30 m / 6 s = -5 m/s.
4. The object’s velocity is 5 m/s during the first 6 seconds and -5 m/s from 6 to 12 seconds. The negative sign indicates movement in the opposite direction.