Name: Equilibrium of a Particle
Uploaded: 2019-07-19
Duration: 10 min 20 s
Description: How do bridges remain static even when subjected to external forces? The principle of equilibrium ensures their structural integrity. Explore the equilibrium...

Question

A flare of mass $0.5 kg$ is fired vertically upwards with speed $30 {ms}^{- 1} .$ The flare itself provides a force of $0.8 N$ upwards, even when the flare is falling, to keep the flare high for as long as possible. The flare is visible over the horizon when it reaches a height of $25 m .$ Find how long the flare is visible for.

Accepted Answer

Given,

Mass of the flare, $m = 0.5 kg$

Initial speed, $u = 30 m s^{- 1}$

Force by the flare, $F = 0.8 N$

The height after which the flare is visible over the horizon is, $s = 25 m$

Suppose the flare is visible for the time duration $T$ .

Take vertically downward upward to be positive.

$F - m g = m a$

where $a$ is acceleration of the flare and $g$ is acceleration due to gravity.

Substituting all the values, we get,

$0.8 - 0.5 \times 10 = 0.5 a$

$\Rightarrow a = \frac{0.8 - 5}{0.5} = - 8.4 m s^{- 2}$

Let $t_{1}$ be the time for which the flare is moving in vertically upward direction, then, using equation,

$v_{1} = u + a t_{1}$

Since at maximum height, $v_{1} = 0_{}$

therefore, $0 = 30 - 8.4 t_{1}$

or, $t_{1} = \frac{30}{8.4} = 3.57 s$

Let $t_{2}$ be the time for which the flare is below horizon and $v_{2}$ be the velocity at $25 m$ .

${v_{2}}^{2} = u^{2} + 2 a s$

Putting all the values, we get,

${v_{2}}^{2} = {(30)}^{2} - 2 \times 8.4 \times 25$

$\Rightarrow v_{2} = \sqrt{480} = 21.9 m s^{- 1}$

Now using the equation, $v_{2} = u + a t_{2}$

$\Rightarrow 21.9 = 30 - 8.4 \times t_{2}$

$\Rightarrow t_{2} = 0.96 s$

Take the difference between both the times,

$t_{1} - t_{2} = 3.57 - 0.96 = 2.61 s$

Total time is calculated for going upward and coming down above the horizon.

So total time taken, $T = 2 \times (t_{1} - t_{2}) = 2 \times 2.61 = 5.22 s$

Hence, the flare will be visible for $5.22 s$ .

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