A particle metes along a straight line.
The fixed point lies on this line.
The displacement of the particle from at time seconds, , is metres where

At time seconds the velocity of is m s where
Find an expression for in terms of :
Give your expression in the form where and are integers to be found.

Ask by Donnelly Colon. in Somalia

Jan 19,2025

Question

A particle metes along a straight line.
The fixed point lies on this line.
The displacement of the particle from at time seconds, , is metres where

At time seconds the velocity of is m s where
Find an expression for in terms of :
Give your expression in the form where and are integers to be found.

Ask by Donnelly Colon. in Somalia

Jan 19,2025

UpStudy AI · Accepted Answer

Let's analyze the problem step by step to find the expression for $$ T $$ in terms of $$ V $$.

### Given:
The displacement $$ s(t) $$ of the particle from the fixed point $$ O $$ at time $$ t $$ seconds is:
$$
s(t) = 5 - t^{3} - 4 t^{2} - 5t - 7
$$
Simplifying:
$$
s(t) = -t^{3} - 4t^{2} - 5t - 2
$$

### Velocity:
The velocity $$ v(t) $$ is the derivative of displacement with respect to time:
$$
v(t) = \frac{ds}{dt} = \frac{d}{dt}\left(-t^{3} - 4t^{2} - 5t - 2\right) = -3t^{2} - 8t - 5
$$
Given that at time $$ T $$, the velocity $$ v(T) = V $$:
$$
-3T^{2} - 8T - 5 = V
$$
Rearranging the equation:
$$
3T^{2} + 8T + (5 + V) = 0
$$

### Solving for $$ T $$:
This is a quadratic equation in terms of $$ T $$. Using the quadratic formula:
$$
T = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}
$$
Here, $$ a = 3 $$, $$ b = 8 $$, and $$ c = 5 + V $$.
$$
T = \frac{-8 \pm \sqrt{64 - 12(5 + V)}}{6} = \frac{-8 \pm \sqrt{64 - 60 - 12V}}{6} = \frac{-8 \pm \sqrt{4 - 12V}}{6}
$$
Simplifying further:
$$
T = \frac{-4 \pm \sqrt{4 - 12V}}{3}
$$
To match the required form $$ \frac{-4 + \sqrt{k + mV}}{3} $$, we select the positive square root:
$$
T = \frac{-4 + \sqrt{4 - 12V}}{3}
$$
Here, $$ k = 4 $$ and $$ m = -12 $$.

### Final Answer:
After simplifying, T is ( –4 + √(4 – 12 V) ) divided by three. Thus,

T = (−4 + √(4 − 12V))/3
$$ T = \frac{-4 + \sqrt{4 - 12V}}{3} $$

A particle metes along a straight line.
The fixed point lies on this line.
The displacement of the particle from at time seconds, , is metres where

At time seconds the velocity of is m s where
Find an expression for in terms of :
Give your expression in the form where and are integers to be found.

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A particle metes along a straight line. The fixed point lies on this line. The displacement of the particle from at time seconds, , is metres where At time seconds the velocity of is m s where Find an expression for in terms of : Give your expression in the form where and are integers to be found.

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A particle metes along a straight line.
The fixed point lies on this line.
The displacement of the particle from at time seconds, , is metres where

At time seconds the velocity of is m s where
Find an expression for in terms of :
Give your expression in the form where and are integers to be found.