DYNAMICS

 

DYNAMICS: SPEED, VELOCITY, AND ACCELERATION

INTRODUCTION

Dynamics is a branch of mechanics that deals with the motion of objects and the forces causing them. It is an essential topic in physics, especially in kinematics, which studies motion without considering the forces behind it. The key concepts in dynamics include speed, velocity, and acceleration. These quantities help in understanding how objects move, change direction, and increase or decrease their motion.

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SPEED

Definition

Speed is the rate at which an object covers a distance. It is a scalar quantity because it has only magnitude and no direction. The SI unit of speed is meters per second (m/s).

Formula

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

Types of Speed

1. Uniform Speed: When an object covers equal distances in equal intervals of time.
2. Variable Speed: When an object covers unequal distances in equal intervals of time.
3. Average Speed: The total distance traveled divided by the total time taken.
4. Instantaneous Speed: The speed of an object at a particular instant.

Example Calculation

A car travels 150 km in 3 hours. Find its speed.

$$\text{Speed} = \frac{150 \text{ km}}{3 \text{ hours}} = 50 \text{ km/h}$$

A simpler way to convert 50 km/h to m/s:

1. Convert km to meters: 50 km = 50,000 meters
2. Convert hours to seconds: 1 hour = 3,600 seconds
3. Divide:

$$\frac{50,000 \text{ m}}{3,600 \text{ s}} = 13.89 \text{ m/s}$$

So, 50 km/h ≈ 13.89 m/s.

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VELOCITY

Definition

Velocity is the rate of displacement of an object. Unlike speed, velocity is a vector quantity because it has both magnitude and direction.

Formula

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

Types of Velocity

1. Uniform Velocity: When an object moves in a straight line with constant speed and direction.
2. Variable Velocity: When an object changes its speed or direction.
3. Average Velocity: The total displacement divided by the total time taken.
4. Instantaneous Velocity: The velocity of an object at a particular instant.

Example Calculation

A train moves 120 m east in 10 s. Find its velocity.

$$\text{Velocity} = \frac{120 \text{ m}}{10 \text{ s}} = 12 \text{ m/s east}$$

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ACCELERATION

Definition

Acceleration is the rate of change of velocity of an object. It is a vector quantity and is measured in meters per second squared (m/s²).

Formula

$$\text{Acceleration} = \frac{\text{Change in velocity}}{\text{Time taken}}$$

or

$$a = \frac{v - u}{t}$$

where:

• $a$ = acceleration
• $v$ = final velocity
• $u$ = initial velocity
• $t$ = time taken

Types of Acceleration

1. Uniform Acceleration: When an object's velocity changes at a constant rate.
2. Variable Acceleration: When an object's velocity changes at different rates.
3. Negative Acceleration (Deceleration): When an object slows down.

Example Calculation

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

$$a = \frac{30 - 10}{5} = \frac{20}{5} = 4 \text{ m/s²}$$

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WORKED EXAMPLES FROM JAMB EXAMINATIONS

Example 1 (JAMB 2015)

A cyclist moves with an average speed of 12 m/s for 20 minutes. How far does he travel?

Solution

$$\text{Distance} = \text{Speed} \times \text{Time}$$

Time = $20 \times 60 = 1200$ s

$$\text{Distance} = 12 \times 1200 = 14,400 \text{ m} = 14.4 \text{ km}$$

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Example 2 (JAMB 2017)

A car moves with a speed of 30 m/s and later accelerates at 5 m/s² for 4 seconds. Find its final velocity.

Solution
Using the formula:

$$v = u + at$$ $$v = 30 + (5 \times 4) = 30 + 20 = 50 \text{ m/s}$$

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Example 3 (JAMB 2019)

An object moving with a velocity of 25 m/s is brought to rest in 5 seconds. Calculate its acceleration.

Solution
Using:

$$a = \frac{v - u}{t}$$ $$a = \frac{0 - 25}{5} = \frac{-25}{5} = -5 \text{ m/s²}$$

(Negative sign indicates deceleration.)

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Example 4 (JAMB 2020)

A vehicle travels 90 m in 3 seconds. What is its speed?

Solution

$$\text{Speed} = \frac{\text{Distance}}{\text{Time}} = \frac{90}{3} = 30 \text{ m/s}$$

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Example 5 (JAMB 2022)

A body accelerates uniformly from rest at 2 m/s² for 10 seconds. Find its final velocity.

Solution
Using:

$$v = u + at$$

Since $u = 0$,

$$v = 0 + (2 \times 10) = 20 \text{ m/s}$$

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JAMB PRACTICE QUESTIONS

Objective Questions

1. A car moves with a speed of 36 km/h. What is its speed in m/s?
   A) 10 m/s
   B) 15 m/s
   C) 20 m/s
   D) 25 m/s

2. A ball rolls down a slope with an initial velocity of 2 m/s and attains a velocity of 10 m/s in 4 seconds. What is its acceleration?
   A) 2 m/s²
   B) 3 m/s²
   C) 4 m/s²
   D) 5 m/s²

3. An object moves in a straight line and covers equal distances in equal intervals of time. This motion is called?
   A) Uniform acceleration
   B) Uniform velocity
   C) Non-uniform acceleration
   D) Variable motion

4. A truck accelerates uniformly from rest at 3 m/s². What will be its speed after 6 seconds?
   A) 12 m/s
   B) 15 m/s
   C) 18 m/s
   D) 21 m/s

5. If a car moves 200 m in 10 s, what is its velocity?
   A) 15 m/s
   B) 20 m/s
   C) 25 m/s
   D) 30 m/s

6. An object moving with uniform velocity has what acceleration?
   A) Zero
   B) 1 m/s²
   C) 5 m/s²
   D) 10 m/s²

7. A car moving at 40 m/s comes to rest in 8 s. What is its acceleration?
   A) -5 m/s²
   B) -6 m/s²
   C) -7 m/s²
   D) -8 m/s²

8. What is the SI unit of acceleration?
   A) m/s
   B) m/s²
   C) kgm/s²
   D) N

9. If a car maintains a speed of 50 m/s for 2 minutes, what distance does it cover?
   A) 2000 m
   B) 3000 m
   C) 4000 m
   D) 6000 m

10. A car moving with a velocity of 30 m/s decelerates at 5 m/s². How long will it take to stop?
    A) 4 s
    B) 5 s
    C) 6 s
    D) 8 s

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This note provides a strong foundation for understanding speed, velocity, and acceleration. Let me know if you need any modifications!

Answers to the questions 

Here are the correct answers with explanations:

1. A car moves with a speed of 36 km/h. What is its speed in m/s?

   • Conversion: $36 \times \frac{1000}{3600} = 10$ m/s
   • Answer: A) 10 m/s

2. A ball rolls down a slope with an initial velocity of 2 m/s and attains a velocity of 10 m/s in 4 seconds. What is its acceleration?

   • Formula: $a = \frac{v - u}{t}$
   • $a = \frac{10 - 2}{4} = \frac{8}{4} = 2$ m/s²
   • Answer: A) 2 m/s²

3. An object moves in a straight line and covers equal distances in equal intervals of time. This motion is called?

   • Definition: This is the definition of uniform velocity.
   • Answer: B) Uniform velocity

4. A truck accelerates uniformly from rest at 3 m/s². What will be its speed after 6 seconds?

   • Formula: $v = u + at$
   • $v = 0 + (3 \times 6) = 18$ m/s
   • Answer: C) 18 m/s

5. If a car moves 200 m in 10 s, what is its velocity?

   • Formula: $v = \frac{\text{distance}}{\text{time}}$
   • $v = \frac{200}{10} = 20$ m/s
   • Answer: B) 20 m/s

6. An object moving with uniform velocity has what acceleration?

   • Definition: Uniform velocity means no change in speed, so acceleration is zero.
   • Answer: A) Zero

7. A car moving at 40 m/s comes to rest in 8 s. What is its acceleration?

   • Formula: $a = \frac{v - u}{t}$
   • $a = \frac{0 - 40}{8} = -5$ m/s²
   • Answer: A) -5 m/s²

8. What is the SI unit of acceleration?

   • Answer: B) m/s²

9. If a car maintains a speed of 50 m/s for 2 minutes, what distance does it cover?

   • Formula: $\text{Distance} = \text{Speed} \times \text{Time}$
   • $50 \times (2 \times 60) = 50 \times 120 = 6000$ m
   • Answer: D) 6000 m

10. A car moving with a velocity of 30 m/s decelerates at 5 m/s². How long will it take to stop?

• Formula: $t = \frac{v - u}{a}$
• $t = \frac{0 - 30}{-5} = 6$ s
• Answer: C) 6 s

Let me know if you need further explanations!

15 Questions on the Three Equations of Motion

1. A car starts from rest and accelerates uniformly at 2 m/s² for 5 seconds. Find its final velocity.
2. A bus is moving at 10 m/s and accelerates uniformly at 3 m/s² for 4 seconds. Find its final velocity.
3. A body starts from rest and moves with an acceleration of 6 m/s² for 8 seconds. How far does it travel?
4. A stone is thrown with a velocity of 20 m/s and moves with an acceleration of 2 m/s² for 3 seconds. Find the distance covered.
5. A train moving at 15 m/s accelerates at 1.5 m/s² for 6 seconds. Find its final velocity and distance travelled.
6. A car slows down uniformly from 30 m/s to 10 m/s in 4 seconds. Find its acceleration.
7. A body moves with an initial velocity of 12 m/s and acceleration of 4 m/s². How long will it take to reach a velocity of 28 m/s?
8. A motorcycle starts from rest and covers 90 m in 6 seconds with uniform acceleration. Find its acceleration.
9. A lorry moving at 25 m/s comes to rest in 5 seconds. Find its retardation.
10. A ball is projected vertically upward with an initial velocity of 18 m/s. Taking acceleration due to gravity as 10 m/s², find the time it takes to reach its highest point.
11. A car moving at 20 m/s accelerates uniformly at 5 m/s² for 3 seconds. Find the distance covered.
12. A body starts from rest and travels 50 m in 5 seconds. Find its acceleration.
13. A cyclist increases speed uniformly from 5 m/s to 15 m/s in 5 seconds. Find the distance covered.
14. A vehicle moving at 40 m/s comes to rest over a distance of 80 m. Find its acceleration.
15. A ball is dropped from a height and reaches the ground in 4 seconds. Taking acceleration due to gravity as 10 m/s², find the height.