Speed Velocity Acceleration Study note

 

1. Speed

Speed is the distance traveled per unit of time. It is a scalar quantity, meaning it only has magnitude (how much) but no direction. Speed does not provide any information about the path or direction of motion.

Key Points about Speed:

• Scalar Quantity: Only tells you how fast something is moving, but not in which direction.
• Constant Speed: When an object covers equal distances in equal intervals of time, its speed is constant.

Formula for Average Speed:

$$\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}}$$

This formula is used when an object’s motion is irregular, i.e., the speed changes over time.

Instantaneous Speed:

This refers to the speed of an object at a specific point in time. It is calculated as:

$$\text{Instantaneous Speed} = \lim_{\Delta t \to 0} \frac{\Delta d}{\Delta t}$$

Where:

• $\Delta d$ is the small change in distance.
• $\Delta t$ is the small change in time.

Instantaneous speed is often what a speedometer in a car shows.

Units:

• The SI unit of speed is meters per second (m/s).
• Other units: kilometers per hour (km/h), miles per hour (mph), feet per second (ft/s).

Example of Speed Calculation:

A cyclist travels 30 kilometers in 2 hours. The speed is:

$$\text{Speed} = \frac{30 \text{ km}}{2 \text{ hours}} = 15 \text{ km/h}$$

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2. Velocity

Velocity is a vector quantity, which means it has both magnitude and direction. It tells you not only how fast an object is moving, but also in which direction it is moving. This makes velocity different from speed, which only tells you how fast an object is moving.

Key Points about Velocity:

• Vector Quantity: It includes both speed and the direction of motion.
• Displacement: The shortest straight-line distance between the starting and ending points, along with the direction of travel.
• Uniform Velocity: When an object moves in a straight line at a constant speed.
• Non-Uniform Velocity: When either the speed or direction (or both) change over time.

Formula for Average Velocity:

$$\text{Average Velocity} = \frac{\text{Displacement}}{\text{Total Time}}$$

Here, displacement refers to the straight-line distance from the initial to the final position, and time is the total duration.

Instantaneous Velocity:

This is the velocity of an object at a specific moment in time. The formula for instantaneous velocity is derived from the concept of instantaneous speed but includes direction:

$$\text{Instantaneous Velocity} = \frac{\Delta \text{Displacement}}{\Delta t}$$

It is a vector, meaning it has both magnitude (speed) and direction.

Units:

• The SI unit of velocity is meters per second (m/s).
• Other units: kilometers per hour (km/h), miles per hour (mph).

Example of Velocity Calculation:

A person walks 30 meters to the east in 10 seconds. The velocity is:

$$\text{Velocity} = \frac{30 \text{ meters east}}{10 \text{ seconds}} = 3 \text{ m/s east}$$

Here, the direction "east" is included, which is what makes this a velocity, not just a speed.

Vector Nature of Velocity:

Velocity’s vector nature means that if you change the direction of motion without changing the speed, the velocity will still change. For example, if you drive around a circular track at a constant speed, your velocity is continually changing because your direction is changing.

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3. Acceleration

Acceleration is a vector quantity that represents the rate of change of velocity over time. It tells you how quickly an object is speeding up or slowing down. It can also refer to changes in direction (in cases like circular motion), not just speed.

Key Points about Acceleration:

• Vector Quantity: Has both magnitude (how much change) and direction (in which direction the velocity is changing).
• Positive Acceleration: Occurs when an object speeds up (increase in velocity).
• Negative Acceleration (Deceleration): Occurs when an object slows down (decrease in velocity).
• Zero Acceleration: Occurs when there is no change in velocity (i.e., the object moves at a constant velocity).

Formula for Acceleration:

$$\text{Acceleration} = \frac{\Delta \text{Velocity}}{\Delta t}$$

Where:

• $\Delta \text{Velocity}$ is the change in velocity (final velocity - initial velocity).
• $\Delta t$ is the time over which the change occurs.

Units:

• The SI unit of acceleration is meters per second squared (m/s²).

Example of Acceleration Calculation:

A car’s velocity increases from 0 m/s to 20 m/s in 5 seconds. The acceleration is:

$$\text{Acceleration} = \frac{20 \text{ m/s} - 0 \text{ m/s}}{5 \text{ seconds}} = 4 \text{ m/s}^2$$

This means the car’s velocity increases by 4 meters per second for every second.

Types of Acceleration:

1. Uniform Acceleration: When an object’s velocity changes by the same amount in each equal time interval (constant acceleration).
   • Example: A car accelerating at a constant rate.
2. Non-Uniform Acceleration: When the rate of change of velocity varies over time.
   • Example: A car that accelerates more rapidly in the beginning and slows down as it approaches its top speed.

Equations of Motion for Constant Acceleration:

If an object is moving with constant acceleration, the following kinematic equations can be used:

1. $v = u + at$

   • $v$ is the final velocity.
   • $u$ is the initial velocity.
   • $a$ is the acceleration.
   • $t$ is the time.

2. $s = ut + \frac{1}{2} a t^2$

   • $s$ is the displacement.
   • $u$ is the initial velocity.
   • $a$ is the acceleration.
   • $t$ is the time.

3. $v^2 = u^2 + 2as$

   • $v$ is the final velocity.
   • $u$ is the initial velocity.
   • $a$ is the acceleration.
   • $s$ is the displacement.

These equations are used when an object is moving under constant acceleration (such as free-falling objects or vehicles accelerating at a steady rate).

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Graphs of Motion

1. Speed-Time (or Velocity-Time) Graph:

   • The slope of the graph represents acceleration.
   • If the graph is a straight horizontal line, the object is moving with constant speed or velocity.
   • If the graph is sloping upward, the object is accelerating.
   • If the graph slopes downward, the object is decelerating (negative acceleration).

2. Displacement-Time Graph:

   • A straight line with a constant slope indicates uniform motion (constant velocity).
   • A curve (increasing slope) indicates acceleration.
   • A downward curve indicates deceleration.

3. Velocity-Time Graph:

   • The area under the curve represents the displacement.
   • A horizontal line represents constant velocity (no acceleration).
   • A straight line with a positive slope represents constant positive acceleration.
   • A straight line with a negative slope represents deceleration.

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Real-World Applications of Speed, Velocity, and Acceleration

1. Driving: Speed limits on roads are based on speed, while navigating requires velocity (as direction matters).
2. Sports: In running, cycling, and swimming, athletes try to maximize their speed, and coaches analyze acceleration for improving performance.
3. Free Fall: Objects in free fall near Earth's surface accelerate at approximately $9.8 \, \text{m/s}^2$ due to gravity, which is an example of constant acceleration.