a particle moving in a straight line moves half a distance with a constant velocity of 10m/s and next half distance with a constant velocity of 5m/s . What is average velocity of particle during the entire motion?
Answers
Let the total distance travelled by the particle be 2 x .
The particle covers half of the distance with constant velocity of 10 m/s and the other half of the distance with constant velocity of 5 m/s .
Half of the distance travelled by the particle will be half of 2 x .
⇒ 1/2 of 2 x ⇒ 1/2 × 2 x ⇒ x
∴ The particle covers x distance with 10 m/s and the remaining x distance with 5 m/s .
Let the total time taken by the particle to cover 2 x distance be T .
The average velocity of the body is the total displacement in unit total time .
Average velocity = 2 x / T .
Let the time taken to cover x with 10 m/s be t₁ and the time taken to cover x with 15 m/s be t₂ .
T = t₁ + t₂
We know that : velocity = displacement / time
⇒ Time = displacement / velocity .
For the average velocity :
⇒ Time T = 2 x / v
⇒ Time T = 2 x / v m/s
For the first case
Velocity = 10 m/s
Time = t₁
Displacement = x
Time t₁ = x / 10 m/s
For the second case
Velocity = 5 m/s
Time = t₂
Displacement = x .
Time t₂ = x / 5 m/s
Average velocity
From the above we can write :
T = t₁ + t₂
⇒ 2 x / v m/s = x / 5 m/s + x / 10 m/s
⇒ 2 x / v m/s = x / m/s ( 1 / 5 + 1 / 10 )
⇒ 2 / v = ( 1 / 5 + 1 / 10 )
⇒ 2 / v = 1 / 5 + 1 / 10
⇒ 2 / v = ( 2 + 1 ) / 10
⇒ 2 / v = 3 / 10
⇒ v = 20 / 3
⇒ v = 6.666 m/s
The average velocity is 6.67 m/s .