Question

A particle travels half the distance of a straight journey with a speed 5 m/s. The remaining part of the distance is covered with speed 6 m/s for half of the remaining time and with speed 4 m/s for the other half of the remaining time. The average speed of the particle is?

Answer 1

so let the total distance be 2x for first half distance  let the time be t₁

so t₁ = x/5  -------------- (i)

so let the total time at second part be t₂

distance covered in first half time is x₁ (let) so x₁ = t₂/2 x 6 ⇒ x₁ = 3t₂

for second half time let the distance be x₂ = t₂/2 x 4 = 2t₂

so we know x₁ + x₂ = x = 5t₂ ⇒ t₂ = x/5  ----------- (ii)

so average speed = total distance / total time = 2x/(t₁ + t₂) = 2x/(x/5 + x/5) 
                                              
                                                                                       = 5m/s ANSWER

Answer 2

Let total distance be 2d.  Let total time taken be t.
let  v = average speed  = 2 d / t

Time to travel the first  "d" =  d/5
Remaining time = t - d/5 = (5 t - d) / 5

Distance covered in the remaining time: 6 (5t-d)/10 + 4 (5t -d)/10
            so  d = 5 t -d

avg speed = v = 2d/t = 5 m/s