Speed and velocity best concept
Today our topic is continue speed and velocity types post number 13 study every post serially for better understand because each post is related with previous post lets start.
Average speed: What is average speed ? why we study average speed ? we know that average speed is define as total distance travelled /total time taken that is average speed = total distance/total time now why we study average speed answer is when we travel distance from one location to other location during this period our speed is changes many times some time larger speed, sometime slow speed so we study average speed during journey.
|Speed and velocity best concept|
Suppose an object travel from point A to B with different speed V₁ for t₁ time, V₂ for t₂ time, V₃ for t₃ time then what will be object average speed figure below.
Speed V₁ V₂ V₃
Time t₁ t₂ t₃
Remember our formula for average speed is given above but here total distance travelled is with different speed that is V₁, V₂, V₃ For this type of problem remember every speed we made section for distance hence in this figure there are three section that is Ap, pq, qB so we know that distance travel with constant speed is distance = speed*time so distance for Ap d₁ = V₁*t₁, distance for pq d₂ = V₂*t₂, distance for qB d₃ = V₃*t₃ hence we can write total distance d = V₁*t₁+V₂*t₂+V₃*t₃ now total time is t = t₁ +t₂ +t₃ now we can write average speed = (V₁*t₁+V₂*t₂+V₃*t₃)/(t₁ +t₂ +t₃)
Now we can write this formula Average speed = ⅀V𝗂*t𝗂/⅀t𝗂 where i limit 1 to n.
now when body moves with different speed for same interval of time then our formula for average speed is equal to
average speed = (V₁t+V₂t+V₃t+………..+V𝑛t)/(t+t+t+t+……………+t𝑛)
average speed = (V₁+V₂+V₃…………………+V𝑛)/n
example a car travel with speed 40 km/hr for one hour and with speed 60 km/hr for next one hour then what is average speed of the car so here time duration for both case is same one hour hence average speed above formula can be used, average speed = (40+60)/2 = 50 km/hr here important is time interval for every speed must be same then we use formula average speed = (V₁+V₂+V₃…………………+V𝑛)/n .
Now for second case same distance with different speed figure below.
Distance S S S
Speed v₁ v₂ v₃
here we see that distance AB, BC, CD are same but object speed is different for each distance so how much of total distance that is total distance = AB+BC+CD = s+s+s = 3s now for general case for n terms will be ns for total time calculation we see here that for every interval speed is different hence time must be different so time for AB distance t₁ = s/v₁, for BC t₂ = s/v₂, for CD t₃ = s/v₃ so total time t = t₁+t₂+t₃ = s/v₁+s/v₂+s/v₃ hence for general n terms
Average speed = ns/(s/v₁+s/v₂+s/v₃…………..+s/v𝓃)
Average Speed = n/(1/v₁+1/v₂+1/v₃+…………+1/v𝓃) ( here s is cancelled taking common ) now rearranging the equation we will get 1/Average speed = 1/n(1/v₁+1/v₂+1/v₃+…………+1/v𝓃) so this formula is when distances are same .
Now take a famous example a car goes half the distance with 40 km/hr and another half distance with 60 km/hr calculate average speed of car. here we can solve this question by two method one is from our general definition average speed or second is our formula for same distance so i am using here formula .
1/Average speed = 1/n(1/v₁+1/v₂) put just value here n =2 because two speed 40 and 60 hence 1/Average speed = 1/2(1/40+1/60) = 1/2(3+2)/120 = 1/48 hence Average speed = 48 km/hr for checking answer is correct or not this average value will lies between lower and higher speed value.A question now for you a car travel half the time with 40 km/hr and another half the time with 60 km/hr find the average speed of car simple solve it using same time formula answer 50 km/hr always remember this types of problem two quantity is given third you have to find.
Q a body travelling along a straight line covered 1/3 rd of total distance with speed 4 m/s remaining part of the distance was covered with speed of 2 m/s for half the time and with 6 m/s other half of time what is average speed for the whole journey. it is type of mixed question but analysis step by step first concept how many times speed changes answer is three times so divide whole journey in three part. figure given below.
s t t
4 m/s 2 m/s 6 m/s
Average Speed = total distance/total time now here total distance is not given so suppose total distance AD = 3s now from question 1/3 rd distance will be s because total distance is 3s here 1/3rd distance speed is 4 m/s so time for this distance t₁= s/4 now rest distance is BD time is equal and speed is different suppose BC time t, and CD time t but speed and distance is different but we know that BD total distance is 2s hence BC distance = 2t, CD distance = 6t therefore
BC+CD = 2s or 2t+6t = 2s or 8t = 2s or t= s/4.
now from equation of average speed = totoal distance/total time.
put the value Average speed = 3s/(t₁+t+t) = 3s/(s/4+s/4+s/4) = 3/3/4= 4 m/s hence Average speed = 4 m/s. so important point is divide section according to how many time speed changes and second point two quantity are given find third quantity.
Now our next point average speed is different from average velocity answer is Yes lets see how it is different.
take an example a body is running on a circle of radius r and complete one revolution in time t.
now here total distance after one revolution = 2𝛑r
total displacement after one revolution = 0 because initial and final point is at same point.
hence average speed = 2𝛑r/t.
Average Velocity = 0/t = 0 hence average speed and average velocity is different important point is when any body turn its direction during motion then its distance and displacement is change then its speed and velocity is different unless it travel in a straight line its distance and displacement will be same then its speed and velocity will be same. hence in a rectilinear motion distance = displacement, speed = velocity take an example .
Q a man walk from home to market 2.5 km away with speed 5 km/hr market is closed he return back immediately with speed 7.5 km/hr find magnitude of average velocity. what is his average speed over 0 to 30 minute, 0 to 50 minute 0 to 40 minute.
figure given below .
Home 2.5 km Market
Speed 5 km/hr
7.5 km/hr ←
Hence from question time taken to reach market = 2.5/5 = .5 hr = 30 minute now returning speed is 7.5 km/hr so time taken to return = 2.5/7.5 = 1/3 = 20 minute now total time to come back home is 30+20 = 50 minute hence average velocity displacement is zero so average velocity is zero.
now 0 to 30 minute average speed = 5 km/hr
now 0 to 50 minute total distance = 2.5+2.5 = 5 km , total time 50 minute = 50/60 for hr hence average speed = 5/5/6 = 6 km/hr now for 0 to 40 minute in returning 10 minute distance travel = 7.5*10/60 = 1.25 km hence total distance = 2.5+1.25 = 3.75 km hence average speed = 3.75/40/60 = 5.625 km/hr.
Q in a clock watch second needle complete one revolution find average velocity and average speed needle length is r also find after 30 second and 45 second .
here displacement after 60 second is zero hence average velocity =0 average speed = 2𝛑r/60 = 𝛑r/30 m/s .
now for 30 second displacement = 2r hence average velocity = 2r/30 = r/15 m/s and average speed = 𝛑r/30 .
now for 45 second displacement = √2 r hence average velocity = √2 r/45 and average speed = 3/4*2𝛑r/45 = 3𝛑r/90 m/s . hence we calculate average speed and average velocity both are different thanks for reading
Speed and velocity best concept
Dated 28th April 2018