For what times is the height of the stone greater than 10m?

Question

A stone is thrown upwards with an initial velocity, v₀, of 30m/s from a height h₀ , of 2m. For what times is the height of the stone greater than 10m? Use height formula h = -16t² + v₀t + h₀ and round the answers to the nearest hundredths of a second.

Answer 1

First, Shaun, clearly define your terms. This will help you and help others who are trying to follow your work.

h: height (meters) of the stone at time t

t: time (seconds) after the stone was thrown

t₀: zero seconds after stone was thrown

v₀: velocity (meters per second) at t₀

h₀: height (meters) at t₀

Formula for the HEIGHT at TIME t: h = —16t² + v₀t + h₀

Fill into the formula the values you are given, leaving out the units.

h = –16t² + 30t + 2

Use your graphing calculator to plot this function or formula: h = –16t² + 30t + 2

Height will be on the vertical axis, and time will be on the horizontal axis. There will be an infinite number of answers, so you'll need to identify the interval where these answers lie by finding where height equals 10. Your answer will be in the following format with the question marks standing for rounded values that you read off your graph: ? ≤ t ≤ ??

Answer 2

h= 10 =-16t^2 + 30t + 2

-16t^2 + 30t -8 = 0

-8t^2 + 15t - 4 =0

t =15/16 + or - (-1/16)sqr(225-128)

t = 15/16 + or - (1/16)sqr97

t = 0.9375 + or -0.6146 = 1.552 or 0.323

the stone will be above 10 meters from t- 0.323 seconds to 1.552 seconds for a total 1.229 seconds

(actually the height equation this problem has is for feet not meters. The deceleration term, -16t^2 is for the effect of gravity in units of feet per second per second. Had this been the force of gravity in meters, the equation would have had -4.9t^2 as the squared term)

h=-16^2 + 30t + 2 take the derivative and set = 0 to find time of max height, time above 10 will be plus or minus the same time for max height.

h' =-32t +30 =0

t = 30/32 =15/16 seconds for maximum height= 0.938 (half way between 1.552 and 0.323)

max height =-/16)^2 + 30(15/16) + 2 =14

Answer 3

h(t) = - 16t² + 30t + 2 > 10, 8t² - 15t + 4 < 0, 8t² - 15t + 4 = 0 t = (15 ± (225 - 128)^1/2)/16, t = 0.322 s or

t = 1.553 s. SO for 0.322 s < t < 1.553 s a stone height more then 10 m