Velocity (College Board AP® Physics 1: Algebra-Based)

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Leander Oates

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Physics

Average velocity

Speed

The speed of an object is defined as:

The distance traveled by an object per unit time

Speed is a scalar quantity with magnitude only
- The direction of the object's motion is not described by speed

Average and instantaneous speed

The speedometer of a car gives a measure of instantaneous speed

For example, if the speedometer reads 50 miles per hour, which is 22 meters per second, it means that the car is traveling 22 meters in each second that this speed is maintained

Close-up of a car's speedometer, showing speeds from 0 to 260 km/h, with glowing red indicators and diesel fuel gauge at half. — **The speedometer on a vehicle displays instantaneous speed**

Image: Creative commons licence from pxhere.com

As the speed of the car changes, the speedometer will give a new value for the new instantaneous speed
- The instantaneous speed is the speed at which an object travels in each instant of time
Average speed, on the other hand, describes the whole journey
Average speed considers the total distance traveled and the total time taken
- For example, the car will not travel at 22 meters per second for its entire journey; it will speed up and slow down accordingly

$average speed = \frac{total distance traveled}{total time taken}$

If the car traveled a total distance of 2000 meters and the whole journey took 3 minutes, which is 180 seconds, then the average speed of the car was 11 meters per second

$average speed = \frac{2000}{180} = 11 m / s$

Map showing distance traveled by car

A hand-drawn map of a town with irregular grid roads and buildings. A red line shows a winding route labeled "Distance traveled." A north arrow is in the top right. — **The red line shows the distance traveled by the car over its 3 minute journey**

Velocity

The velocity of an object is defined as:

The displacement of an object per unit time

In other words, the rate of change of an object's position
Velocity is a vector quantity with both magnitude and direction
- If an object travels with a constant speed but changes direction, then its velocity is changing
- Therefore, it is possible for an object to travel at a constant speed without a constant velocity, but it is not possible for an object to travel at a constant velocity without a constant speed
The magnitude of an object's velocity is its speed
- However, the magnitude of an object's average velocity is not its average speed

Average and instantaneous velocity

Average velocity also describes the whole journey of an object
Average velocity considers the initial and final states of an object over an interval of time
- In other words, the displacement of an object over the total time taken

${\vec{v}}_{a v g} = \frac{∆ \vec{x}}{∆ t}$

Where:
- ${\vec{v}}_{a v g}$ = average velocity, measured in $m / s$
- $∆ \vec{x}$ = displacement, measured in $m$
- $∆ t$ = time interval, measured in $s$
If the car had a displacement of 1500 meters over its 3 minute journey, then its average velocity would be 8.3 meters per second travelling north-east for that same car trip

${\vec{v}}_{a v g} = \frac{1500}{180} = 8.3 m / s NE$

Map showing displacement of car

Map showing a red winding path for distance travelled and a straight blue line for displacement. Includes a key and data: distance 2000m, displacement 1500m NE. — **Map showing distance traveled in red and displacement in blue. Since these values are different, the average speed and the average velocities are also different.**

Calculating the average velocity over a very small time interval yields a value that is very close to the instantaneous velocity

Negative velocity

Since velocity is a vector quantity, it can have a positive or negative value
A positive velocity value indicates an object is traveling in the initial direction of motion
A negative value of velocity indicates that an object is traveling in the opposite direction
If there is no initial direction of motion, then positive velocity is generally either:
- forwards
- to the right
- upwards
However, mathematically, any direction can be assigned as positive as long as the opposing direction is negative

Examiner Tip

Whichever direction you assign as positive, make sure you are consistent throughout your calculation.

Worked Example

The map below shows a high school teacher's 4 minute route through the city on their way to work. The teacher travels 1.8 km along a straight road, then turns left and travels a further 450 m before parking their car.

City street map with a red arrow pointing upwards along a major vertical road heading north, then turning a corner onto another street heading east. A compass indicating north is in the upper-left corner.

What is the ratio of the teacher's average speed to average velocity?

A: 1.1

B: 1.2

C: 1.3

D: 1.4

The correct answer is B

Answer:

Step 1: Analyze the scenario

The roads that the teacher traveled on are perpendicular
- The length of the displacement vectors on the northbound and eastbound roads are known
- These vectors can be added using trigonometry to give the resultant displacement vector

The same city map shows the resultant displacement vector forming the hypotenuse of a triangle with the north and east-bound vectors.

The average velocity of the teacher can then be calculated using the displacement and the time interval of the journey
The total distance traveled is the non-vector sum of the lengths of the displacement vectors on the northbound and eastbound roads

Step 2: List the known quantities

Displacement on northbound road, $x_{N} = 1.8 km = 1800 m$
Displacement on eastbound road, $x_{E} = 450 m$
Time interval, $∆ t = 4 \min = 4 \cdot 60 s = 240 s$

Step 3: Determine the displacement

Using trigonometry, where $c$ is the hypotenuse and therefore $∆ x$

$c^{2} = a^{2} + b^{2}$

$∆ x^{2} = {x_{N}}^{2} + {x_{E}}^{2}$

$∆ x = \sqrt{1800^{2} + 450^{2}}$

$∆ x = 1855 m NE$

Step 4: Calculate the average velocity

${\vec{v}}_{a v g} = \frac{1855}{240} = 7.73 m / s NE$

Step 5: Calculate the average speed

Determine the total distance traveled

$total distance = 1800 + 450 = 2250 m$

Determine the average speed

$average speed = \frac{total distance traveled}{total time taken}$

$average speed = \frac{2250}{240} = 9.38 m / s$

Step 6: Calculate the ratio of average speed to average velocity

$\frac{average speed}{average velocity} = \frac{9.38}{7.73} = 1.21$

$1.21 = 1.2 (2 s . f .)$

This is answer B

Examiner Tip

The gap between the question and the answer can seem huge at first glance. It is important that you do not panic when you first read a question. All the information you require to answer the question will always be provided for you; you just need to think logically about how you can use that information to get the quantities you actually want.

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Velocity (College Board AP® Physics 1: Algebra-Based)

Study Guide

Average velocity

Speed

Average and instantaneous speed

Map showing distance traveled by car

Velocity

Average and instantaneous velocity

Map showing displacement of car

Negative velocity

Examiner Tip

Worked Example

Examiner Tip

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Author: Leander Oates