Question

Please really need help with question d really don't understand

The answer to the question is the second image, I don't get how they got this answer and why it isn't finding x for t=0, t=1 and t=2.

Answer 1

`\textsf{a)} \quad v=12t^2-6t-18`

`\textsf{b)} \quad a=24t-6`

c) 1.5 s

d) -19.25 m

e) 24.5 m

Explanation:

Displacement

`x=4t^3-3t^2-18t+1`

(where t ≥ 0 and x is in meters)

Part (a)

To find the equation for velocity, differentiate the equation for displacement:

`\implies v=\frac{\text{d}x}{\text{d}t}=12t^2-6t-18`

(where t ≥ 0 and v is in meters per second)

Part (b)

To find the equation for acceleration, differentiate the equation for velocity:

`\implies a=\frac{\text{d}v}{\text{d}t}=24t-6`

(where t ≥ 0 and a is in meters per second squared)

Part (c)

The particle comes to rest when its velocity is zero:

`\begin{aligned}v & = 0\\\implies 12t^2-6t-18 & = 0\\6(2t^2-t-3) & = 0\\2t^2-t-3 & = 0\\2t^2-3t+2t-3 & = 0\\t(2t-3)+1(2t-3) & = 0\\(t+1)(2t-3) & = 0\\\implies t & =-1, (3)/(2)\end{aligned}`

As t ≥ 0, the particle comes to rest at 1.5 s.

Part (d)

Substitute the found value of t from part (c) into the equation for displacement to find where the particle comes to rest:

`\implies 4(1.5)^3-3(1.5)^2-18(1.5)+1=-19.25\: \sf m`

Part (e)

We have determined that the particle is at rest at 1.5 s.

Therefore, to find how far the particle traveled in the first 2 seconds, we need to divide the journey into two parts: before and after it was at rest.

The first leg of the journey is the first 1.5 s and the second leg of the journey is the next 0.5 s.

At the beginning of the journey, t = 0 s.

`\textsf{when }t=0: \quad x=4(0)^3-3(0)^2-18(0)+1=1`

Therefore, when t = 0, x = 1

When t = 1.5 s, x = -19.25 m (from part (d)).

⇒ Total distance traveled in first 1.5 s = 1 + 19.25 = 20.25 m

When t = 2, x = -15 m.

Therefore, the particle has traveled 19.25 - 15 = 4.25 m in the last 0.5 s of its journey.

Total distance traveled:

1 + 19.25 + 4.25 = 24.5 m

Refer to the attached diagram

When the particle is at rest, it changes direction. If we model its journey using the x-axis, for the first leg of its journey (0 - 1.5 s) it travels in the negative direction (to the left). At 1.5 s it stops, then changes direction and travels in the positive direction (to the right), arriving at -15 at 2 seconds.