Speed, Average Velocity and Instantaneous Velocity, Grade 10 Physics

Definition: Velocity

Velocity is the rate of change of position.​

Definition: Instantaneous velocity

Instantaneous velocity is the velocity of an accelerating body at a specific instant in time.​

Definition: Average velocity

Average velocity is the total displacement of a body over a time interval.​

Velocity is the rate of change of position. It tells us how much an object’s position changes in time. This is the same as the displacement divided by the time taken. Since displacement is a vector and time taken is a scalar, velocity is also a vector. We use the symbol v for velocity. If we have a displacement of ∆x and a time taken of ∆t, v is then defined as:

\[v = \dfrac{\Delta x}{\Delta t}\]​

Velocity can be positive or negative. Positive values of velocity mean that the object is moving
away from the reference point or origin and negative values mean that the object is moving
towards the reference point or origin.

Important: An instant in time is different from the time taken or the time interval. It is therefore useful to use the symbol t for an instant in time (for example during the 4th second) and the symbol ∆t for the time taken (for example during the first 5 seconds of the motion).

Average velocity (symbol v) is the displacement for the whole motion divided by the time taken for the whole motion. Instantaneous velocity is the velocity at a specific instant in time. (Average) Speed (symbol s) is the distance travelled (d) divided by the time taken (∆t) for the journey. Distance and time are scalars and therefore speed will also be a scalar. Speed is calculated as follows:

\[s = \dfrac{d}{\Delta t}\]​

Instantaneous speed is the magnitude of instantaneous velocity. It has the same value, but no direction.
Worked Example 1: Average speed and average velocity
Question: James walks 2 km away from home in 30 minutes. He then turns around and walks back home along the same path, also in 30 minutes. Calculate James’ average speed and average velocity.

Hướng dẫn

Step 1 : Identify what information is given and what is asked for
The question explicitly gives

• the distance and time out (2 km in 30 minutes)
• the distance and time back (2 km in 30 minutes)

Step 2 : Check that all units are SI units.
The information is not in SI units and must therefore be converted.
To convert km to m, we know that:
2 km = 2 000 m
Similarly, to convert 30 minutes to seconds
30 min = 1 800 s
Step 3 : Determine James’ displacement and distance.
James started at home and returned home, so his displacement is 0 m
∆x = 0 m
James walked a total distance of 4 000 m (2 000 m out and 2 000 m back).
d = 4 000 m
Step 4 : Determine his total time.
James took 1 800 s to walk out and 1 800 s to walk back.
∆t = 3 600 s
Step 5 : Determine his average speed

Step 6 : Determine his average velocity

Worked Example 2: Instantaneous Speed and Velocity
Question: A man runs around a circular track of radius 100 m. It takes him 120 s to complete a revolution of the track. If he runs at constant speed, calculate:

1. his speed,
2. his instantaneous velocity at point A,
3. his instantaneous velocity at point B,
4. his average velocity between points A and B,
5. his average speed during a revolution.
6. his average velocity during a revolution.

Hướng dẫn

Step 1 : Decide how to approach the problem
To determine the man’s speed we need to know the distance he travels and how long it takes. We know it takes 120 s to complete one revolution of the track.(A revolution is to go around the track once.)
Step 2 : Determine the distance travelled
What distance is one revolution of the track? We know the track is a circle and we know its radius, so we can determine the distance around the circle. We start with the equation for the circumference of a circle
C = 2πr = 2π(100 m) = 628,32 m
Therefore, the distance the man covers in one revolution is 628,32 m.
Step 3 : Determine the speed
We know that speed is distance covered per unit time. So if we divide the distance covered by the time it took we will know how much distance was covered for every unit of time. No direction is used here because speed is a scalar.
\[s = \dfrac{d}{\Delta t}\] = 5,24m∙s^-1
Step 4 : Determine the instantaneous velocity at A
Consider the point A in the diagram. We know which way the man is running around the track and we know his speed. His velocity at point A will be his speed (the magnitude of the velocity) plus his direction of motion (the direction of his velocity). The instant that he arrives at A he is moving as indicated in the diagram. His velocity will be 5,24 m·s$^{−1 }$West.
Step 5 : Determine the instantaneous velocity at B
Consider the point B in the diagram. We know which way the man is running around the track and we know his speed. His velocity at point B will be his speed (the magnitude of the velocity) plus his direction of motion (the direction of his velocity). The instant that he arrives at B he is moving as indicated in the diagram. His velocity will be 5,24 m·s$^{−1}$ South.
Step 6 : Determine the average velocity between A and B
To determine the average velocity between A and B, we need the change in displacement between A and B and the change in time between A and B. The displacement from A and B can be calculated by using the Theorem of Pythagoras:

Δx² = 100² + 100² => Δx = 100√2 m
The time for a full revolution is 120 s, therefore the time for a1/4 of a revolution is 30 s.
v$_{AB}$ = \[\dfrac{\Delta x}{\Delta t}\] = 4,71s
Velocity is a vector and needs a direction. Triangle AOB is isosceles and therefore angle BAO = 45◦ .The direction is between west and south and is therefore southwest. The final answer is: v = 4.71 m·s$^{−1}$, southwest.
Step 7 : Determine his average speed during a revolution
Because he runs at a constant rate, we know that his speed anywhere around the track will be the same. His average speed is 5,24 m·s$^{−1}$.
Step 8 : Determine his average velocity over a complete revolution
Important: Remember - displacement can be zero even when distance travelled is not! To calculate average velocity we need his total displacement and his total time. His displacement is zero because he ends up where he started. His time is 120 s. Using these we can calculate his average velocity:

\[v = \dfrac{\Delta x}{\Delta t}\] = 0/120 = 0​

2/ Differences between Speed and Velocity
The differences between speed and velocity can be summarised as:

Additionally, an object that makes a round trip, i.e. travels away from its starting point and then returns to the same point has zero velocity but travels a non-zero speed.

Exercise: Displacement and related quantities
E-1: Theresa has to walk to the shop to buy some milk. After walking 100 m, she realises that she does not have enough money, and goes back home. If it took her two minutes to leave and come back, calculate the following:
(a) How long was she out of the house (the time interval ∆t in seconds)?
(b) How far did she walk (distance (d))?
(c) What was her displacement (∆x)?
(d) What was her average velocity (in m·s$^{−1}$)?
(e) What was her average speed (in m·s$^{−1}$)?
E-2: Desmond is watching a straight stretch of road from his classroom window. He can see two poles which he earlier measured to be 50 m apart. Using his stopwatch, Desmond notices that it takes 3s for most cars to travel from the one pole to the other.
(a) Using the equation for velocity (v =∆x/∆t), show all the working needed to calculate the velocity of a car travelling from the left to the right.
(b) If Desmond measures the velocity of a red Golf to be -16,67 m·s$^{−1}$, in which direction was the Gold travelling? Desmond leaves his stopwatch running, and notices that at t = 5,0 s, a taxi passes the left pole at the same time as a bus passes the right pole. At time t = 7,5 s the taxi passes the right pole. At time t = 9,0 s, the bus passes the left pole.
(c) How long did it take the taxi and the bus to travel the distance between the poles? (Calculate the time interval (∆t) for both the taxi and the bus).
(d) What was the velocity of the taxi and the bus?
(e) What was the speed of the taxi and the bus?
(f) What was the speed of taxi and the bus in km·h$^{−1}$?
E-3: After a long day, a tired man decides not to use the pedestrian bridge to cross over a freeway, and decides instead to run across. He sees a car 100 m away travelling towards him, and is confident that he can cross in time.
(a) If the car is travelling at 120 km·h$^{−1}$, what is the car’s speed in m·s$^{−1}$.
(b) How long will it take the a car to travel 100 m?
(c) If the man is running at 10 km·h$^{−1}$, what is his speed in m·s$^{−1}$?
(d) If the freeway has 3 lanes, and each lane is 3 m wide, how long will it take for the man to cross all three lanes?
(e) If the car is travelling in the furthermost lane from the man, will he be able to cross all 3 lanes of the freeway safely?

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