Speed and Velocity – SS1 Physics Lesson Note (First Term)

Speed is the distance covered per unit time, while velocity is the displacement per unit time in a stated direction.

Behavioural objectives

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

1. define speed, average speed and velocity, and state which are scalar and which are vector quantities;
2. define acceleration and retardation;
3. write the equations of uniformly accelerated motion and state what each symbol stands for;
4. solve simple problems on speed and uniformly accelerated motion;
5. state the information that can be obtained from a velocity-time graph.

What are scalar and vector quantities?

A scalar quantity has magnitude (size) only, while a vector quantity has both magnitude and direction.

• Scalar quantities: distance, speed, time, mass and energy.
• Vector quantities: displacement, velocity, acceleration and force.
• Distance: the total length of the path travelled by a body, in any direction.
• Displacement: the distance moved in a stated direction, measured in a straight line from the starting point.

What is speed?

Speed is the rate of change of distance moved, or the distance covered per unit time, in no specified direction.

• Quantity type: scalar.
• Unit: metres per second (m/s).

Speed = distance moved ÷ time

What is average speed?

Average speed is the ratio of the total distance travelled to the total time taken.

• Quantity type: scalar.
• Unit: m/s or m s^-1.

Average speed = total distance travelled ÷ total time taken

What is uniform speed?

Uniform speed (constant speed) is the motion of a body that covers equal distances in equal time intervals, no matter how small the time interval may be.

• Distance-time graph: a straight sloping line.
• Example: a car that covers 20 m every second along a straight road.

Worked example: average speed

A runner covers 400 m in 50 s. Find the average speed.

Solution:

Average speed = total distance ÷ total time
= 400 m ÷ 50 s
= 8 m/s

What is velocity?

Velocity is the rate of change of displacement, that is, the distance moved in a specific direction per unit time.

• Quantity type: vector.
• Unit: metres per second (m/s).
• Example: a car travelling at a steady speed of 50 km/h in the direction N40°E has a velocity of 50 km/h, N40°E.

Velocity = displacement ÷ time

What is uniform (constant) velocity?

An object has uniform (constant) velocity if the rate of change of its displacement is constant, no matter how small the interval may be.

• Meaning: both the speed and the direction stay the same.
• Velocity-time graph: a horizontal straight line.

What is the difference between speed and velocity?

Speed is distance per unit time with no direction, while velocity is displacement per unit time with direction.

How do you convert km/h to m/s?

To convert km/h to m/s, multiply by 1000 and divide by 3600, which is the same as multiplying by 5/18.

1. Change kilometres to metres (1 km = 1000 m).
2. Change hours to seconds (1 h = 3600 s).
3. Divide the metres by the seconds.

• Example: 54 km/h = 54 × 5/18 = 15 m/s.

Worked example: distance from speed and time

A train moves at a speed of 54 km/h for one-quarter minute. Find the distance travelled by the train.

Solution:

Speed = 54 km/h = 15 m/s

Time = ¼ min = ¼ × 60 = 15 s

Distance = speed (m/s) × time (s)
= 15 (m/s) × 15 (s)
= 225 m

What are acceleration and retardation?

Acceleration is the rate of increase of velocity, and retardation is the rate of decrease of velocity.

What is acceleration?

Acceleration is the rate of increase of velocity, measured in m/s².

Acceleration (a) = increase in velocity ÷ time taken

• Uniform acceleration: the velocity of a moving body increases by an equal amount in equal intervals of time, no matter how small the intervals may be.
• Quantity type: vector.

What is retardation?

Retardation is the rate of decrease of velocity, measured in m/s².

Retardation (a_r) = decrease in velocity ÷ time taken

• Other names: deceleration and negative acceleration.
• Example: a bus slowing down as it approaches a bus stop.

Worked example: acceleration

A car increases its velocity from 10 m/s to 30 m/s in 5 s. Find its acceleration.

Solution:

Increase in velocity = 30 − 10 = 20 m/s
a = 20 ÷ 5
= 4 m/s²

Worked example: retardation

A bus slows from 20 m/s to 5 m/s in 5 s. Find its retardation.

Solution:

Decrease in velocity = 20 − 5 = 15 m/s
Retardation = 15 ÷ 5
= 3 m/s²

What are the equations of uniformly accelerated motion?

The equations of uniformly accelerated motion are four equations that link u, v, a, S and t for a body moving with constant acceleration.

1. S = (v + u)t ÷ 2
2. v = u + at
3. v² = u² + 2aS
4. S = ut + ½at²

Equations 1 to 4 are used to solve problems on uniformly accelerated motion, where:

• u: initial velocity (m/s).
• v: final velocity (m/s).
• a: acceleration (m/s²).
• S: distance covered (m).
• t: time (s).

Which equation should you use?

Choose the equation that contains the three quantities you know and the one you want, and leaves out the quantity you do not need.

How do you solve a problem on uniformly accelerated motion?

1. Write down the quantities given, using the symbols u, v, a, S and t.
2. If the body starts from rest, write u = 0.
3. For retardation, use a negative value for a.
4. Choose the equation that fits the quantities.
5. Substitute the values and solve, then write the unit.

Worked example: final velocity from acceleration and distance

A car moves from rest with an acceleration of 0.2 m/s². Find its velocity when it has moved a distance of 50 m.

Solution:

a = 0.2 m/s², S = 50 m, u = 0 m/s, v = ?

v² = u² + 2aS
v² = 0² + (2 × 0.2 × 50) = 20
v = √20 m/s
v ≈ 4.5 m/s

Worked example: distance from acceleration and time

A car starts from rest and accelerates at 2 m/s² for 5 s. Find the distance covered.

Solution:

u = 0 m/s, a = 2 m/s², t = 5 s, S = ?

S = ut + ½at²
S = (0 × 5) + (½ × 2 × 5²)
S = 25 m

How are motion graphs used?

The motion of an object is best described with graphs.

1. Distance-time graph: the slope gives the speed.
2. Displacement-time graph: the slope gives the velocity.
3. Velocity-time graph: the slope gives the acceleration, and the area under it gives the distance.

What does a velocity-time graph show?

A velocity-time graph shows how the velocity of an object changes with time.

The velocity-time graph is more useful than the other two graphs because it gives more information about the motion of objects.

• Acceleration: the slope of a line that rises (part A).
• Retardation: the slope of a line that falls (part C).
• Uniform velocity: a horizontal line (part B).
• Distance: the area between the graph line and the time axis.
• Average speed: total distance (total area) ÷ total time.

How do you find distance from a velocity-time graph?

The distance moved equals the area under the velocity-time graph.

1. Split the area under the graph into simple shapes such as a square, rectangle, triangle or trapezium.
2. Find the area of each shape.
3. Add the areas together to get the total distance moved.

• Area of a rectangle: length × breadth.
• Area of a triangle: ½ × base × height.
• Area of a trapezium: ½ × (sum of parallel sides) × height.

Worked example: distance from a velocity-time graph

A body accelerates uniformly from rest to 10 m/s in 4 s, moves at 10 m/s for 6 s, then comes to rest in 2 s. Find the total distance and the average speed.

Solution:

Distance in the first 4 s = ½ × 4 × 10 = 20 m
Distance in the next 6 s = 6 × 10 = 60 m
Distance in the last 2 s = ½ × 2 × 10 = 10 m
Total distance = 20 + 60 + 10 = 90 m

Total time = 4 + 6 + 2 = 12 s
Average speed = 90 ÷ 12 = 7.5 m/s

Evaluation

1. Define speed and state whether it is a scalar or a vector quantity.
2. State the difference between speed and velocity.
3. What is retardation? Give two other names for it.
4. Write the four equations of uniformly accelerated motion.
5. List the information that can be obtained from a velocity-time graph.

Assignment

Explain why the velocity-time graph is more useful than the distance-time and displacement-time graphs, and show how the distance moved by an object is obtained from its velocity-time graph.