A particle moves 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 im s where
Find an expression for in terms of :
Give your expression in the form where and are integers to be found.

Ask by Clarke Wang. in Somalia

Jan 19,2025

Question

A particle moves 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 im s where
Find an expression for in terms of :
Give your expression in the form where and are integers to be found.

Ask by Clarke Wang. in Somalia

Jan 19,2025

UpStudy AI · Accepted Answer

Let's carefully analyze the problem to derive the required expression for $$ T $$ in terms of $$ V $$.

### Given:
- **Displacement Function:** $$ s(t) = t^3 + 4t^2 - 5t - 7 $$
- **Velocity Function:** $$ v(t) = \frac{ds}{dt} = 3t^2 + 8t - 5 $$

### Condition:
At time $$ T $$ seconds, the velocity $$ v(T) = V $$. Therefore, we set:
$$ 3T^2 + 8T - 5 = V $$

### Solving for $$ T $$:
Rearrange the equation to form a quadratic in terms of $$ T $$:
$$ 3T^2 + 8T - (5 + V) = 0 $$

Using the quadratic formula $$ T = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$, where $$ a = 3 $$, $$ b = 8 $$, and $$ c = -(5 + V) $$:
$$
T = \frac{-8 \pm \sqrt{64 + 60 + 12V}}{6} \
T = \frac{-8 \pm \sqrt{124 + 12V}}{6} \
$$

Simplify the square root:
$$
\sqrt{124 + 12V} = \sqrt{4(31 + 3V)} = 2\sqrt{31 + 3V}
$$

Substitute back:
$$
T = \frac{-8 \pm 2\sqrt{31 + 3V}}{6} = \frac{-4 \pm \sqrt{31 + 3V}}{3}
$$

Since $$ T \geq -5 $$, we take the positive root:
$$
T = \frac{-4 + \sqrt{31 + 3V}}{3}
$$

### Conclusion:
The expression for $$ T $$ in terms of $$ V $$ is:
$$
T = \frac{-4 + \sqrt{31 + 3V}}{3}
$$
where $$ k = 31 $$ and $$ m = 3 $$.

**Final Answer:**
After simplifying, T is ( –4 + √(31 + 3 V) ) divided by three. Thus,

T = (−4 + √(31+3V))/3

where k = 31 and m = 3.
$$ T = \frac{-4 + \sqrt{31 + 3V}}{3} $$, where $$ k = 31 $$ and $$ m = 3 $$.

A particle moves 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 im 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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