The velocity V m/s of a particle in motion is given by 𝑉 = 3𝑡

Question

The velocity V m/s of a particle in motion is given by 𝑉 = 3𝑡

2 − 2𝑡 + 5.Calculate the distance
travelled by the particle between t= 2 seconds and t = 6 seconds. (3 marks)

Answer 1

To calculate the distance traveled by the particle, we need to find the displacement function by integrating the velocity function.

The displacement function is the integral of the velocity function:

𝑊(𝑡) = ∫ (3𝑡^2 − 2𝑡 + 5) 𝑑𝑡

Evaluating the integral gives us:

𝑊(𝑡) = 𝑡^3 − 𝑡^2 + 5𝑡 + 𝐶

To find the constant of integration, we use the initial condition 𝑊(2) = 0, since the particle starts at t = 2 seconds.

0 = (2)^3 − (2)^2 + 5(2) + 𝐶
0 = 8 - 4 + 10 + 𝐶
𝐶 = -14

So, the displacement function becomes:

𝑊(𝑡) = 𝑡^3 − 𝑡^2 + 5𝑡 - 14

To find the distance traveled, we need to find the absolute value of the displacement between t = 2 seconds and t = 6 seconds:

Distance = |𝑊(6) - 𝑊(2)|

Distance = |(6)^3 - (6)^2 + 5(6) - 14 - [(2)^3 - (2)^2 + 5(2) - 14]|

Distance = |216 - 36 + 30 - 14 - 8 + 4 - 10 + 14|

Distance = |206|

Therefore, the distance traveled by the particle between t = 2 seconds and t = 6 seconds is 206 meters.

Answer 2

To calculate the distance traveled by the particle between t=2 seconds and t=6 seconds, we need to find the displacement first.

The displacement, Δx, can be obtained by integrating the velocity function over the given interval:

Δx = ∫(V dt) = ∫((3t^2 - 2t + 5) dt)

To evaluate this integral, we need to separate the terms and integrate them individually:

Δx = ∫(3t^2 dt) - ∫(2t dt) + ∫(5 dt)

Integrating each term:

Δx = t^3 + t^2 + 5t + C

Now, to find the distance traveled, we need to find the absolute value of the displacement between t=2 seconds and t=6 seconds:

Distance = |Δx(t=6) - Δx(t=2)|

Substituting the values:

Distance = |(6^3 + 6^2 + 5(6)) - (2^3 + 2^2 + 5(2))|

Calculating:

Distance = |(216 + 36 + 30) - (8 + 4 + 10)|

Distance = |282 - 22|

Distance = |260|

Therefore, the distance traveled by the particle between t=2 seconds and t=6 seconds is 260 meters.