The height in metres of a stone thrown by a student t seconds after it is thrown into the air is given by H(t)=-5t^2+30t+2a) Explain why it is -5^2 and not 5^2 b) The time taken to reach its...

The height in metres of a stone thrown by a student t seconds after it is thrown into the air is given by H(t)=-5t^2+30t+2

a) Explain why it is -5^2 and not 5^2

b) The time taken to reach its maximum height

c) The maximum height of the stone

d) How long before it hits the ground?

e) From what height was the stone thrown?

Asked on by mr-bean

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hala718's profile pic

hala718 | High School Teacher | (Level 1) Educator Emeritus

Posted on

H(t) = -5t^2 +30t + 2

a) The factor for t^2 is negative because the rock was thrown against the gravity. If the rock was falling toward gravity, then the factor will be positive.

b)To reach maximum height. We need to find the maximum value of H(t):

First we need to determine first derivative's zeros.

H(t) = -5t^2 + 30 t + 2

==> H'(t) = -10t + 30 = 0

==> t= 3

B) Then the height will be at maximum point when the time t= 3

C) To find the maximum height we will substitute with t=3.

Then the maximum hight is:

H(3) = -5*3^2 + 30*3 + 2

= -45 + 90 + 2 = 47

Then the maximum height is 47 .

d) The time before it hits the ground. This means we need the time when the height is 0. (hit the ground)

So, we will substitute with h= 0 and determine t:

Note that there will be two values for t, the first when first we through the rock, and the second will be when it hit the ground.

H(t) = -5t^2 + 30t + 2

==>-5t^2 + 30 t + 2 = 0

==> t1= [-30 + sqrt(900-4*-5+2)]/2*-5

= [-30 + sqrt(940)]/-10

= (-30+ 30.7)/-10

= (-0.7/-10 = 0.07

==> t1= 0.07 ( thisis te initial time when the roch first thrown.

==> t2= (-30 -30.7)/10 = -60.7/-10 = 6.07

Then the time needed for the roch to hit the ground is:

t= 6.07 seconds.

 

e) The height where the och thrown is:

H(t1) = H(0.07) = 4.0755

neela's profile pic

neela | High School Teacher | (Level 3) Valedictorian

Posted on

The formula for the height reached by a stone thrown assumes an initial velocity u (upwards) and the gravitational acceleration g downwards. So the height reached in t seconds is : Integral (u -gt) dt = ut-(1/2)t^2 +C .  So H(t) = ut-(1/2)gt^2+C.

a)

Therefore  comparing -5t^2 + 30t +2 and ut -(1/2)gt^2+C , -5 and -(1/2)g are  similar. The gravitational acceleration  and velocity are opposite -5t^2 is correct.

b) H(t) = -5t^2+30t+2 = 2+45 - 5(t- 3)^2 = 47 - 5(t-3)^2.

So when t -3 = 0  H(t) = 47 is highest. t -3 = 0 when t = 3.

c) Already obtained in (b): highest H(t) = 47.

d) To attain  attain a height of 47= (2+45)  and a similar fall , it takes 6seconds and  to fall the initial height you require a futher time t given by  2 = 5t^2+30t . So  t = {-30 +sqrt[30^2-4*5*(-2)]}/2*5 = (-30+30.24903)/10 = 0.06594 secs

Therefore time to reach the ground = 6+0.066 seconds = 6.066 secs.

e) The  initial height is obtained by setting t= 0 in H(t) = -5t^2+30+2. So H(0) = -5*0^2+30*0+2 = 2. So 2 is the initial height from which the stone is thrown is 2.

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