Using the quadratic formula

I have been asked to calculate the maximum height a ball can achieve after being lauched, using;

h = - 4.9t^2 + 19.6t - 12.6

I knew straight away I required the time, so I did the following;

t = -b + or - square root b^2 - 4ac / 2a

t = __- 19.6 + or - square root 19.6^2 - 4 (- 4.9)(- 12.6)__

2 x ( - 4.9)

t = __- 19.6 + or - square root 384.16 - 246.96__

2 x ( - 4.9)

t = __- 19.6 + or - square root 137.20__

2 x( - 9.80)

t = __- 19.6 + or - 11.71__

- 9.80

t = __- 7.89__

- 9.80

t1 = 0.81

t2 = __- 19.6 + or - 11.71__

- 9.80

t2 = __- 31.31__

- 9.80

t2 = 3.20

I then used both routes to conclude the height in metres to determine if there was any significant difference, there wasn't so I will use t2;

h = - 4.9 x 3.20^2 + 19.6 x 3.20 - 12.6

h = - 50.18 + 62.72 - 12.6

h = 0

This is the third answer I have had using the above formula?

I have had first 65m, then 34m and now 0 metres?

I can see I must have tapped a interger in wrong in the calculator or not used it correctly, but can I ask, does the above now look like I have got it right?

Thanks

David(Wondering)

Re: Using the quadratic formula

There's been asked the maximum height the ball can achieve, you're calculating the time wherefore the h=0 and so when the ball reach the ground.

Use:

Now you're calculating after how much time the ball will reach his highest point and then you've just to enter this time in the equation.

Re: Using the quadratic formula

Quote:

Originally Posted by

**Siron** There's been asked the maximum height the ball can achieve, you're calculating the time wherefore the h=0 and so when the ball reach the ground.

Use:

Now you're calculating after how much time the ball will reach his highest point and then you've just to enter this time

in the equation.

Hi Siron, so are you saying that the height above the cliff is 7m using the equation you quote?

I see what you mean now, the equation you use is taking the average of the two routes of the t1 and t2 I found, I never thought about averaging the routes?

Thanks

David

Re: Using the quadratic formula

Yes, the maximum height a ball can achieve is indeed 7m en that's you wanted to calculate. If you want to calculate after how much time the ball will reach the ground and so h=0, then you've indeed to use the quadratic formula.

Re: Using the quadratic formula

Quote:

Originally Posted by

**Siron** Yes, the maximum height a ball can achieve is indeed 7m en that's you wanted to calculate. If you want to calculate after how much time the ball will reach the ground and so h=0, then you've indeed to use the quadratic formula.

Thanks for your help, much appreciated.

Re: Using the quadratic formula

Re: Using the quadratic formula

Or you can use the completing the square method to find the maximum height and time taken to reach maximum height

h = - 4.9t^2 + 19.6t - 12.6

= -4.9 (t^2 - 4t) - 12.6

= -4.9 (t^2 - 4t + 4 - 4 ) - 12.6 Add and subtract square of half the coefficient of linear term to complete the square.

= -4.9(t - 2)^2 + 19.6 - 12.6

h = -4.9(t - 2)^2 + 7

Hence, the maximum height is 7 and after 2 second, which can be found from the vertex (2, 7) of the parabola.

Re: Using the quadratic formula

True, there are a lot of methods to calculate the maximum/minimum of a parabola. Calculating the first derivative and then is also a possibilty (but David hasn't learned derivatives yet).