Average speed: Type 1 : When speed and distances are given. Case a: When unequal portion of distances are travelled at varying speeds: av. When each of values of distance portions are given: Average speed = (Total distance / Total time taken) When a vehicle travels a certain distance dl km at a speed of Sikm/hr and a certain distance d2 km at a speed of S2 km/hr, the average speed is given by, S = (dl+d2)/ ((d1/S1) + (d2/S2)) Example: When a car travels first 60km at the rate of 40km/hr and the next 80km at the rate of 70km/hr. What is its average speed? In this case, we will directly apply equation (1). Hence, average speed,S = 52.973km/hr. a2: When all different distance portions are expressed as a distance ratio of a single portion: When a vehicle travels a certain distance dl km at a speed of Sl km/hr and another distance (x*dl) at a speed of S2 km/hr, the average speed is given by, (2). Example: When a car travels a certain distance at 50 km/hr and (1/3) of the initial distance at 90 km/hr, what is its average speed? Its average speed S= ((1 +(1 (applying equation 2). = 56.25 km/hr. Case b: When equal fractions of distances are travelled at varying speeds: When the vehicle travels (1/2) of the distance at a speed of Sl km/hr and the remaining (1/2) of the distance at a speed of S2 km/hr , equation (1) becomes 2d1/((d1/S1)+(d1/S2)) (3) Similarly, when the vehicle travels (1/3) of the distance at a speed of Sl km/hr,second and third (1/3) of the distances at speeds S2 and S3 km/hr respectively, Average speed = (4) In general, when a vehicle travels (I/N) of the distance at a speed of Sl m/hr and the subsequent (N-l) fractions each of a distance of (1 [N) at varying speeds of .....SN km/hr
2
respectively, Average speed: of similar sequence of( N-2) terms)). Example: When a vehicle travels first 100 km at a speed of 60 km/hr and the next 100 km at a speed of 80 km/hr, Its average speed = (2*60*80)/(60+80)= 68.57 km/hr (by applying equation 3). Type 2: When speed and time are given: Case a: When ,for unequal time periods, varying speeds are encountered: al: when the value of each of the time periods are specified: When a vehicle travels at a speed of Sl km/hr for a time period of tl hours and at a speed of S2 km/hr for a time period of t2 hours, Average speed, S = ((t1*S1)+(t2*S2))/(t1+t2) Example : When a car travels first 4 hours at a speed of 70 km/hr and the next 1.5 hours at a speed of 100 km/hr, what is its average speed? Its average speed ,S= ((4*70)+(1.5*100))/(4+1.5) (applying equation 5). = 78.1818 km/hr. u: When all different time periods are expressed as a ratio of single time period: When a vehicle travels at a speed of Sl km/hr for a time period of ti hours and at a speed of S2 km/hr for a time period of (x*tl) hours, Average speed, S= ((t1*S1)+((x*t1)*S2))/(t1+(x*t1)) (6) Example: When a car travels at a speed of 40 km/hr for a certain time period and at a speed of 60 km/hr for half the initial time period, what is its average speed? Its average speed, S= (40+(0.5*60))/1.5 (applying equation 6). = 46.66 km/hr. Case b: When, for equal time periods, varying speeds are encountered: When a vehicle travels for first ti hours at a speed of Sl km/hr and the next ti hours at a speed of S2 km/hr, Average speed, Arithmetic average (7) Example: When a car travels at a speed of 30 km/hr for the first 2 hours and at a speed of 50 km/hr for the next 2 hours, what is its average speed?
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Its average speed= (30+50)/2 (applying equation 7)= 40 km/hr.
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