if a ball is thrown vertically upward with a velocity of 98ft/s then it's height after t seconds is s=98t-7t^2

Question

if a ball is thrown vertically upward with a velocity of 98ft/sec then it's height after t seconds is s=98t-7t^2.What is the maximmum height reached by the ball?

Answer 1

At the maximum height, the ball has to stop in mid-air and start falling back down to the earth. Take the derivative of the position function:

s' = v = 98 - 14t

This is the velocity function. If you set it equal to zero (i.e. when the ball reaches maximum height), you'll get the time when it reaches maximum height.

0 = 98 - 14t
98 = 14t
t = 7 seconds

Use t = 7 in the position function to find maximum height:

s = 98*(7) - 7*(7)^2
=686 - 343
=343 ft

Answer 2

Velocity at any time (v) = ds/dt

s = 98 t - 7 t^2
v = ds/dt = 98 - 14 t

Maximum distance is where the velocity equals zero
put v =0, then

0 = 98 - 14 t ==> t = 14

substitute for t = 14 in the distance equation

s = 98 (14) - 7 (7)^2 = 343 ft

Answer 3

Using Calculus:
Take the derivative and set it equal to 0.
s' = 98 - 14t
98 - 14t = 0
14t = 98
t = 98/14
t = 7
Replace t with 7 in the function s.
s(7) = 98(7) - 7(7^2)
= 686 - 343
= 343

Answer 4

s=98t-7t^2 = -7(t^2-14+49)+49•7 =-7(t-7)^2+343 (completing the square).

This is maximum when (t-7)^2=0 or when t=7. and at this point the height is 343 ft.

Yes, this of course can also be done with calculus, but I figure you only need to use calculus when you need to. If you don't agree, please see the answers below, they have also done a good job.

Answer 5

Where is this ball being thrown, Mars? Here on Earth our acceleration due to gravity is a little more than 14 ft/s^2 - in fact it's 32 ft/s^2.

Using the proper value for Earth's gravity at sea level gives us a maximum height of 150 ft.

Answer 6

interior the formula of kinematics v^2 = 2as the position v is initial velocity, a is the equipment acceleration and s is distance. fixing for s we've s = v^2/2a on your situation v = 40 8 and a = 32 , so s = 2304/sixty 4 = 36 ft.