a train travels the first 30 km of 120 km track with a uniform speed of 30km/h.What should be the speed of train to cover the remaining distance of the track so that its average speed is 60km/h for the entire trip
To solve this problem, we'll use the formula for average speed:
Average Speed = Total Distance / Total Time
We know that the total distance is 120 km, and we need to find the speed for the remaining distance. Let's denote the remaining distance as D km.
First, let's find the time it takes to travel the first 30 km with a speed of 30 km/h. We can use the formula:
Time = Distance / Speed
So, the time taken to travel the first 30 km is:
Time1 = 30 km / 30 km/h = 1 hour
Now, let's find the time it takes to travel the remaining distance (D km) with an unknown speed (let's call the speed v km/h). Again, we can use the formula for time:
Time2 = D km / v km/h
Now, we know that the average speed for the entire trip is 60 km/h. So, we can write the equation for average speed:
60 km/h = (30 km + D km) / (1 hour + Time2)
Simplifying this equation:
60 km/h = (30 km + D km) / (1 + D/v) hour
Cross-multiplying:
60 km/h * (1 + D/v) hour = 30 km + D km
60 km + 60D/v km/h * hour = 30 km + D km
60D/v km/h * hour - D km = 30 km - 60 km
D (60/v - 1) km = -30 km
D (60 - v)/v km = -30 km
Now, let's solve for the remaining distance D:
D = -30 km * v / (60 - v)
Since distance cannot be negative, we'll discard the negative sign:
D = 30 km * v / (60 - v)
So, the speed of the train to cover the remaining distance should be v km/h, where v is given by:
v = D * (60 - v) / 30 km
Now, we can solve this equation to find the value of v.
Total distance = 120 km
For an average of 60km/h,
total time available = 120/60=2 hours
Distance left = 120-30=90 km
Time left = 2-1 = 1 hour
What is the speed need for the remaining distance?
avg speed= distance/totaltime
total time=time1+time2
= 30km/30km/hr + time2
avg speed=60=120/total time
total time=120/60=2h
30km/30km/hr + time2=2 hr
time2=1hr
but speed2=90km/1hr=90 km/hr