Speed and Types of Speed concept with solved Numerical

 

Speed and Types of Speed concept with solved Numerical

Target: CBSE Board Exams | State Board Exams | NTSE | Olympiads | Foundation 

Speed:

• Definition: Speed is the rate at which an object covers a certain distance.
• Tells us : It tells us about how fast an object is moving
• Quantity : It is a Scalar Quantity (gives only magnitude but it does not tell us about direction)
• SI unit : its SI unit is metre per second (m/s)
• Formula:

                                                            Speed = Distance / Time

                                        Where,                

                                                                Distance – Total path covered (in meters)

                                                                Time – Time taken (in seconds)

    Other units of speed:

• Km/h (Kilometre per hour)
• Cm/s (centimeter per second)

            To convert:

                                         

Types of Speed:

1) Uniform Speed :

Definition : An object is said to be moving with uniform speed if it covers equal distances in equal intervals of time, no matter how small the time interval is.

Example :

•       A car moving on a straight highway at 60 km/h
•       Earth revolving around the Sun (Approximately uniform speed)

Graphical Representation:

• If we plot distance vs time graph for uniform speed, it is a straight line because the distance increases linearly with time.

2) Non-uniform Speed :

Definition : An object is said to be moving with non-uniform speed if it covers unequal distances in equal intervals of time. In other words, the speed keeps changing with time.

Example :

• A car moving in heavy traffic (sometimes fast, sometimes slow).
• A ball rolling down a slope (speed increases gradually).

Graphical Representation:

• For non-uniform speed, the distance vs. time graph is a cured line, since the distance does not increase proportionally with time.

Key Differences between Uniform and Non-uniform Speed:

Feature	Uniform Speed	Non-Uniform Speed
Definition	Equal distance in equal time intervals	Unequal distance in equal time intervals
Nature	Constant	Variable
Graph	Straight line	Curved line (Non-straight line)
Example	Car at 60 km/h on highway	Car in city traffic

3) Average Speed :

Definition : When an object moves with non-uniform speed, we cannot describe its motion with a single constant speed. Instead, we use average speed, which is defined as:

                     Average Speed = Total Distance Traveled / Total Time Taken

Key Points:

• It is different from arithmetic mean of speeds.
• Average speed depends on total distance and total time, not on individual speeds alone.
• It is a scalar quantity (no direction).

Formula of Average Speed :

1) General Formula :

                    Average speed (v_avg) = (d₁ + d₂ + d₃ + …)/ (t₁ + t₂ + t₃ + …)

 2) When distances are equal but speeds are different (say v₁  and v₂) :

                    Average speed (v_avg) = (2 v₁ v₂) / (v₁ + v₂)

Solved Numerical on Average Speed :

Numerical 1 – Basic

A car travels 100 km in 2 hours and then 200 km in 4 hours. Find the average speed.

Solution:

            Total Distance = 100 + 200 = 300 km
            Total Time = 2 + 4 = 6 h
                                          v_avg  = 300 / 6 = 50 km/h

Numerical 2 – Different Speeds, Equal Distances

A bus travels 60 km at 30 km/h and then 60 km at 60 km/h. Find the average speed.

Solution:
            Time for first 60 km = 60/30 = 2 h
            Time for second 60 km = 60/60 = 1 h
            Total Distance = 120 km
            Total Time = 3 h
                                        v_avg = 120/3 = 40 km/h

Numerical 3 – Equal Distance Formula

A car goes from A to B with 40 km/h and returns with 60 km/h. Find the average speed.

Solution:
            v_avg  = (2 v1 v2) / (v1 + v2)

                     = (2 × 40 × 60) / (40 + 60)
                     = 4800/100 
                     = 48 km/h

Numerical 4 – Conversion Required

A person walks 300 m in 3 minutes, then 500 m in 5 minutes. Find the average speed in m/s.

Solution:
            Total Distance = 300 + 500 = 800 m
            Total Time = (3+5) min = 480 s
                                    v_avg = 800/480 = 1.67 m/s

Numerical 5 – Train Problem

A train covers the first 120 km at 60 km/h and the next 180 km at 90 km/h. Find the average speed.

Solution:
            Time for 120 km = 120/60 = 2 h
            Time for 180 km = 180/90 = 2 h
            Total Distance = 300 km
            Total Time = 4 h
                                    v_avg  = 300/4 = 75 km/h

4) Instantaneous Speed :

Definition : Instantaneous Speed is the speed of a moving object at a particular instant of time.

• Key Points : It is what we see on a speedometer of a car or bike at any given moment.
•  Mathematically, it is the limit of average speed when the time interval becomes very small (approaches zero).

Formula :

                                                 v_insta = lim(Δt→0) (Δd/Δt)

Examples in Daily Life :

·         Reading speed from a car speedometer.
·         A sprinter running faster at the end of the race than at the start.
·         A train leaving a station slowly but later moving fast.

Solved Numerical on Instantaneous Speed : 

Numerical 1 – Basic

A car covers 10 metres in 0.5 seconds at a certain instant. Find its instantaneous speed.

Solution:
                                         v_insta = d/t = 10/0.5 = 20 m/s

Numerical 2 – Small Interval

A bike covers 1.2 m in 0.05 s. What is its instantaneous speed?

Solution:
                                         v_insta = 1.2/0.05 = 24 m/s

Numerical 3 – Using Velocity Equation

A car’s displacement is given by equation: s = 5t² + 2t (s in m, t in s). Find the instantaneous speed at t = 4 s.

Solution:
                                         v_insta = ds/dt

                                        v_insta  = d(5t² + 2t)/dt

                                        v_insta  = 10t + 2
                         At t=4:

                                               V_t=4 = 10×4 + 2

                                               V_t=4 = 42 m/s

Numerical 4 – Physics Application

A particle moves such that its displacement is s = 4t³ (s in m, t in s). Find the instantaneous speed at t = 2 s.

Solution:
                                         v_insta = ds/dt

                                         v_insta  = d(4t³)/dt

                                          v_insta = 12t²
                         At t=2:

                                         v_t=2 = 12×(2²)

                                        v_t=2  = 48 m/s

Numerical 5 – Advanced

A car moves such that its displacement is given by: s = 3t² + 2t + 5. Find instantaneous speed at t = 6 s.

Solution:
                                         v_insta = ds/dt

                                        v_insta = d(3t² + 2t + 5)/dt

                                        v_insta = 6t + 2
                         At t=6:

                                             v_t=6 = 6×6 + 2

                                             v_t=6  = 38 m/s

Key Difference between Average Speed and Instantaneous Speed :

Feature	Average Speed	Instantaneous Speed
Definition	Total distance / total time	Speed at a given instant
Interval	Large interval of time	Extremely small time interval
Example	60 km/h for whole journey	65 km/h shown by car’s speedometer at some moment
Nature	Overall motion	Specific moment motion

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