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Speed, Average Velocity and Instantaneous Velocity

Definition 1: Velocity

Velocity is the rate of change of displacement.

Definition 2: Instantaneous velocity

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

Definition 3: 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 vv for velocity. If we have a displacement of ΔxΔx and a time taken of ΔtΔt, vv is then defined as:

velocity ( in m · s - 1 ) = change in displacement ( in m ) change in time ( in s ) v = Δ x Δ t velocity ( in m · s - 1 ) = change in displacement ( in m ) change in time ( in s ) v = Δ x Δ t
(1)

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.

Tip:

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

Average velocity (symbol vv) 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 ss) is the distance travelled (dd) divided by the time taken (ΔtΔt) for the journey. Distance and time are scalars and therefore speed will also be a scalar. Speed is calculated as follows:

speed ( in m · s - 1 ) = distance ( in m ) time ( in s ) speed ( in m · s - 1 ) = distance ( in m ) time ( in s )
(2)
s = d Δ t s = d Δ t
(3)

Instantaneous speed is the magnitude of instantaneous velocity. It has the same value, but no direction.

Exercise 1: Average speed and average velocity

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.

Figure 1
Figure 1 (PG10C2_009.png)

Solution

  1. 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)
  2. 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:

    1 km = 1 000 m 2 km = 2 000 m ( multiply both sides by 2 , because we want to convert 2 km to m . ) 1 km = 1 000 m 2 km = 2 000 m ( multiply both sides by 2 , because we want to convert 2 km to m . )
    (4)

    Similarly, to convert 30 minutes to seconds,

    1 min = 60 s 30 min = 1 800 s ( multiply both sides by 30 ) 1 min = 60 s 30 min = 1 800 s ( multiply both sides by 30 )
    (5)
  3. Step 3. Determine James' displacement and distance. :

    James started at home and returned home, so his displacement is 0 m.

    Δ x = 0 m Δ x = 0 m
    (6)

    James walked a total distance of 4 000 m (2 000 m out and 2 000 m back).

    d = 4 000 m d = 4 000 m
    (7)
  4. Step 4. Determine his total time. :

    James took 1 800 s to walk out and 1 800 s to walk back.

    Δ t = 3 600 s Δ t = 3 600 s
    (8)
  5. Step 5. Determine his average speed :
    s = d Δ t = 4 000 m 3 600 s = 1 , 11 m · s - 1 s = d Δ t = 4 000 m 3 600 s = 1 , 11 m · s - 1
    (9)
  6. Step 6. Determine his average velocity :
    v = Δ x Δ t = 0 m 3 600 s = 0 m · s - 1 v = Δ x Δ t = 0 m 3 600 s = 0 m · s - 1
    (10)

Exercise 2: Instantaneous Speed and Velocity

A man runs around a circular track of radius 100m100m. It takes him 120s120s 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.
Figure 2
Figure 2 (PG10C2_010.png)

Solution

  1. 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.)

  2. 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 = 2 π r = 2 π ( 100 m ) = 628 , 32 m C = 2 π r = 2 π ( 100 m ) = 628 , 32 m
    (11)

    Therefore, the distance the man covers in one revolution is 628,32m628,32m.

  3. 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=dΔt=628,32m120s=5,24m·s-1s=dΔt=628,32m120s=5,24m·s-1
    (12)

  4. 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 5,24m·s-15,24m·s-1 West.

    Figure 3
    Figure 3 (PG10C2_011.png)
    Figure 4
    Figure 4 (PG10C2_012.png)

  5. 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 5,24m·s-15,24m·s-1 South.

    Figure 5
    Figure 5 (PG10C2_013.png)
    Figure 6
    Figure 6 (PG10C2_014.png)

  6. 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:

    (Δx)2=(100m)2+(100m)2=20000mΔx=141,42135...m(Δx)2=(100m)2+(100m)2=20000mΔx=141,42135...m
    (13)
    The time for a full revolution is 120 s, therefore the time for a 1414 of a revolution is 30 s.
    vAB=ΔxΔt=141,42...m30s=4.71m·s-1vAB=ΔxΔt=141,42...m30s=4.71m·s-1
    (14)
    Figure 7
    Figure 7 (PG10C2_015.png)
    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=4.71m·s-1v=4.71m·s-1, southwest.

  7. 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 5,24m·s-15,24m·s-1 .

  8. Step 8. Determine his average velocity over a complete revolution :
    Tip:
    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 120s120s. Using these we can calculate his average velocity:

    v = Δ x Δ t = 0 m 120 s = 0 m · s - 1 v = Δ x Δ t = 0 m 120 s = 0 m · s - 1
    (15)

Differences between Speed and Velocity

The differences between speed and velocity can be summarised as:

Table 1
Speed Velocity
1. depends on the path taken 1. independent of path taken
2. always positive 2. can be positive or negative
3. is a scalar 3. is a vector
4. no dependence on direction and so is only positive 4. direction can be guessed from the sign (i.e. positive or negative)

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.

Displacement and related quantities

  1. Theresa has to walk to the shop to buy some milk. After walking 100m100m, 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:
    1. How long was she out of the house (the time interval ΔtΔt in seconds)?
    2. How far did she walk (distance (d))?
    3. What was her displacement (ΔxΔx)?
    4. What was her average velocity (in m··s-1-1)?
    5. What was her average speed (in m··s-1-1)?
    Figure 8
    Figure 8 (PG10C2_016.png)
    Figure 9
    Figure 9 (PG10C2_017.png)
    Click here for the solution
  2. Desmond is watching a straight stretch of road from his classroom window. He can see two poles which he earlier measured to be 50m50m apart. Using his stopwatch, Desmond notices that it takes 3s3s for most cars to travel from the one pole to the other.
    1. Using the equation for velocity (vv = ΔxΔtΔxΔt), show all the working needed to calculate the velocity of a car travelling from the left to the right.
    2. If Desmond measures the velocity of a red Golf to be -16,67m·s-1-16,67m·s-1, in which direction was the Gold travelling? Desmond leaves his stopwatch running, and notices that at t=5,0st=5,0s, a taxi passes the left pole at the same time as a bus passes the right pole. At time t=7,5st=7,5s the taxi passes the right pole. At time t=9,0st=9,0s, the bus passes the left pole.
    3. How long did it take the taxi and the bus to travel the distance between the poles? (Calculate the time interval (ΔtΔt) for both the taxi and the bus).
    4. What was the velocity of the taxi and the bus?
    5. What was the speed of the taxi and the bus?
    6. What was the speed of taxi and the bus in km·h-1km·h-1?
    Figure 10
    Figure 10 (PG10C2_018.png)
    Click here for the solution
  3. A rabbit runs across a freeway. There is a car, 100m100m away travelling towards the rabbit.
    Figure 11
    Figure 11 (PG10C2_019.png)
    1. If the car is travelling at 120km·h-1120km·h-1, what is the car's speed in m·s-1m·s-1.
    2. How long will it take the a car to travel 100m100m?
    3. If the rabbit is running at 10km·h-110km·h-1, what is its speed in m·s-1m·s-1?
    4. If the freeway has 3 lanes, and each lane is 3m3m wide, how long will it take for the rabbit to cross all three lanes?
    5. If the car is travelling in the furthermost lane from the rabbit, will the rabbit be able to cross all 3 lanes of the freeway safely?
    Click here for the solution

Investigation : An Exercise in Safety

Divide into groups of 4 and perform the following investigation. Each group will be performing the same investigation, but the aim for each group will be different.

  1. Choose an aim for your investigation from the following list and formulate a hypothesis:
    • Do cars travel at the correct speed limit?
    • Is is safe to cross the road outside of a pedestrian crossing?
    • Does the colour of your car determine the speed you are travelling at?
    • Any other relevant question that you would like to investigate.
  2. On a road that you often cross, measure out 50m50m along a straight section, far away from traffic lights or intersections.
  3. Use a stopwatch to record the time each of 20 cars take to travel the 50m50m section you measured.
  4. Design a table to represent your results. Use the results to answer the question posed in the aim of the investigation. You might need to do some more measurements for your investigation. Plan in your group what else needs to be done.
  5. Complete any additional measurements and write up your investigation under the following headings:
    • Aim and Hypothesis
    • Apparatus
    • Method
    • Results
    • Discussion
    • Conclusion
  6. Answer the following questions:
    1. How many cars took less than 3s3s to travel 50m50m?
    2. What was the shortest time a car took to travel 50m50m?
    3. What was the average time taken by the 20 cars?
    4. What was the average speed of the 20 cars?
    5. Convert the average speed to km·h-1km·h-1.

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