Question #248e7

1 Answer
Aug 23, 2017

v_"final" = 20.8 "m/s"

Explanation:

We're asked to find the speed of the ball as it reaches the ground, given its initial height and initial velocity.

We're given that its initial height is 12.2 "m", and that it was kicked solely horizontally, so there is no initial y-velocity.

We know the horizontal speed remains ideally unchanged throughout the motion, so we need to find the final y-velocity of the ball when it hits the ground.

To do so, we can use the equation

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

where

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

  • v_(0y) is the initial y-velocity (0 because the initial velocity was horizontal)

  • g = 9.81 "m/s"

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

  • y_0 is the initial position (12.2 "m")

Plugging in known values:

(v_y)^2 = 0 - 2(9.81color(white)(l)"m/s"^2)(0-12.2color(white)(l)"m")

v_y = +-sqrt(239color(white)(l)"m"^2"/s"^2) = color(red)(15.5color(white)(l)"m/s"

Since we're dealing with speed and not velocity, I neglected the sign, because speed is not a vector quantity.

Now, we use the Pythagoream theorem to find the final speed given the components:

  • v_x = 13.9 "m/s" (x-motion doesn't change in idealized projectile motion)

  • v_y = color(red)(15.5color(white)(l)"m/s"

And so we have

color(blue)(v) = sqrt((v_x)^2 + (v_y)^2) = sqrt((13.9color(white)(l)"m/s")^2 + (color(red)(15.5color(white)(l)"m/s"))^2) = color(blue)(ulbar(|stackrel(" ")(" "20.8color(white)(l)"m/s"" ")|)

which agrees with the answer you gave us.