Name: Normals to the Parabolas
Uploaded: 2019-07-19
Duration: 9 min 41 s
Description: This video will give a clear understanding of how to write and learn the concept of normal to the parabola in the parametric form. Don't miss it.

Question

A child's toy launches toy rockets into the air. The height,

h

m , of a toy rocket which is launched into the air with an initial velocity of

v_{0}

and initial height of

h_{0}

can be modelled by the function,

h (t) = h_{0} + v_{0} (t) - 4.9 t^{2}

, where

t

is time in seconds that have passed since the rocket was launched. A toy rocket is launched from the ground with an initial velocity of

50 m / s

. Find the maximum height the rocket reaches, and the time that passes before it hits the ground again.

Accepted Answer

A child's toy launches toy rockets into the air. The height, $h$ m , of a toy rocket which is launched into the air with an initial velocity of $v_{0}$ and initial height of $h_{0}$ can be modelled by the function, $h (t) = h_{0} + v_{0} (t) - 4.9 t^{2}$ , where $t$ is time in seconds that have passed since the rocket was launched.

If the rocket hits the ground. That means when the rocket's height $h (t) = 0$ and the initial height $h_{0} = 0$

Therefore, $h_{0} + v_{0} (t) - 4.9 t^{2} = 0$

$\Rightarrow 50 (t) - 4.9 t^{2} = 0$ [Initial velocity $v_{0} = 50 m / s$ ]

$\Rightarrow t (50 - 4.9 t) = 0$

$\Rightarrow t = 0 or t = \frac{50}{4.9}$

$\Rightarrow t = 0 or t = 10.2$ seconds

That means that at height $= h (t) = 0, t = 0 or 10.2 . At t = 0$ zero seconds have gone by and the rocket hasn't yet left the ground, so $t = 10.2$ is when the rocket will return to the ground. That means the rocket will take $10.2$ seconds to return to the ground.

Now, differentiate with respect to $t$ , then we get: $\frac{d h}{d t} = 50 - 9.8 t$

For maxima, we have; $\frac{d h}{d t} = 50 - 9.8 t = 0$

$\Rightarrow 50 = 9.8 t$

$\Rightarrow t = 5.10$ seconds

Hence, the maximum height at $t = 5.10$ seconds $h (5.10) = 0 + 50 (5.10) - 4.9 {(5.10)}^{2} = 127.6$ m.

Important Questions on Measuring Change: Differentiation

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