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
Introduction:
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 :
2) When distances are equal but speeds are different (say v1 and v2) :
Average speed (vavg) = (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
vavg = 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
vavg = 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:
vavg = (2 v1 v2) / (v1 + v2)
= 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
vavg = 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
vavg = 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
:
vinsta = lim(Δt→0) (Δd/Δt)
Examples in Daily Life :
· Reading speed from a car speedometer.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:
vinsta = 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:
vinsta = 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:
vinsta = ds/dt
vinsta = d(5t² + 2t)/dt
vinsta = 10t + 2
At t=4:
Vt=4 = 10×4 + 2
Vt=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:
vinsta = ds/dt
vinsta = d(4t³)/dt
vinsta = 12t²
At t=2:
vt=2 = 12×(2²)
vt=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:
vinsta = ds/dt
vinsta =
d(3t² + 2t + 5)/dt
vinsta =
6t + 2
At t=6:
vt=6 = 6×6 + 2
vt=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 |
Conclusion:
अंत में, गति और उसके विभिन्न प्रकारों की समझ भौतिकी के अध्ययन के लिए अत्यंत महत्वपूर्ण है। सॉल्व्ड न्यूमेरिकल्स के अभ्यास से छात्रों की गणना करने की क्षमता, गति और सटीकता में सुधार होता है। नियमित अभ्यास और स्पष्ट अवधारणाओं के साथ छात्र इस विषय में अच्छी पकड़ बना सकते हैं और परीक्षा में बेहतर प्रदर्शन कर सकते हैं।
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