Question #c287b

1 Answer
Aug 23, 2017

(A) "max height" = 87.9 "m"

(B) "final velocity" = -41.5 "m/s"

(C) "time" = 0.767 "s"

Explanation:

We're asked to find

  • (A) the maximum height the stone reaches

  • (B) the velocity at which the stone strikes the ground

  • (C) the time taken to reach the maximum height

" "

  • (A) Maximum Height

To find the maximum height obtained by the stone (and all other information we need), we'll first find the initial velocity of the stone, using the equation

ul(y = y_0 + v_(0y)t - 1/2g t^2

where

  • y is the final position (0, ground level)

  • y_0 is the initial position (85 "m")

  • v_(0y) is the initial velocity (what we're trying to find)

  • t is the time (5 "s")

  • g = 9.81 "m/s"^2

Plugging in known values:

0 = 85 "m" + v_(0y)(5color(white)(l)"s") - 1/2(9.81color(white)(l)"m/s"^2)(5color(white)(l)"s")^2

1/2(9.81color(white)(l)"m/s"^2)(5color(white)(l)"s")^2 = 85color(white)(l)"m" + v_(0y)(5color(white)(l)"s")

color(red)(v_(0y)) = (1/2(9.81color(white)(l)"m/s"^2)(5color(white)(l)"s")^2 - 85color(white)(l)"m")/(5color(white)(l)"s") = color(red)(ul(7.53color(white)(l)"m/s"

At the stone's maximum height, its instantaneous y-velocity is zero, so we can use the equation

ul((v_y)^2 = (v_(0y))^2 - 2g(y - y_0)

The variables here we know except for the final height y, which will be

0 = (color(red)(7.53color(white)(l)"m/s"))^2 - 2(9.81color(white)(l)"m/s"^2)(y - 85color(white)(l)"m")

color(blue)(ulbar(|stackrel(" ")(" "y = 87.9color(white)(l)"m"" ")|)

" "

(B) Final Velocity

To find the final velocity, we can use the equation

ul(v_y = v_(0y) - g t

where

  • v_y is the final velocity (what we're trying to find)

  • v_(0y) is the initial velocity (found to be color(red)(7.53color(white)(l)"m/s")

  • g = 9.81 "m/s"^2

  • t = 5 "s"

Plugging these in:

color(blue)(v_y) = color(red)(7.53color(white)(l)"m/s") - (9.81color(white)(l)"m/s"^2)(5color(white)(l)"s") = color(blue)(ulbar(|stackrel(" ")(" "-41.5color(white)(l)"m/s"" ")|)

" "

  • (C) Time at Maximum Height

To find the time when the stone reaches its maximum height, we can again use the equation

ul(v_y = v_(0y) - g t

where

  • v_y = 0 (at maximum height)

  • v_(0y) = color(red)(ul(7.53color(white)(l)"m/s"

  • g = 9.81 "m/s"

  • t is the time (what we're trying to find)

And so we have

0 = color(red)(7.53color(white)(l)"m/s") - (9.81color(white)(l)"m/s"^2)t

color(blue)(ulbar(|stackrel(" ")(" "t = 0.767color(white)(l)"s"" ")|)