Question #2027f

1 Answer
Aug 15, 2017

(i) 7.967.96 "m"m

(ii) 0.240.24 "s"s

(iii) 0.3750.375 "s"s and 2.172.17 "s"s

Explanation:

We're asked to find

  • (i) the maximum height obtained by the mass

  • (ii) the time it takes the mass to travel 3.03.0 "m"m horizontally

  • (iii) the time(s) when the height of the mass is 4.04.0 "m"m

(i)

Finding the maximum height:

When the particle is at its maximum height, the instantaneous yy-velocity is 00, and we can use the kinematics equation

ul((v_y)^2 = (v_0sinalpha_0)^2 - 2g(Deltay)

where

  • v_y is the y-velocity at height Deltay (which is 0)

  • v_0 is the initial speed (given as 25 "m/s")

  • alpha_0 is the launch angle (given as 30^"o")

  • g = 9.81 "m/s"^2

  • Deltay is the change in height (what we're trying to find)

Plugging in known values:

0 = [(25color(white)(l)"m/s")sin(30^"o")]^2 - 2(9.81color(white)(l)"m/s"^2)(Deltay)

color(red)(ulbar(|stackrel(" ")(" "Deltay = 7.96color(white)(l)"m"" ")|)

The maximum height is thus color(red)(7.96color(white)(l)"meters".

" "

(ii)

The time it takes to travel 3.0 "m" horizontally:

To do this, we can use the equation

ul(Deltax = v_0cosalpha_0t

where

  • Deltax is the change in horizontal position (3.0 "m")

  • v_0 = 25 "m/s"

  • alpha_0 = 30^"o"

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

Plugging in known values:

3.0color(white)(l)"m" = (25color(white)(l)"m/s")cos(30^"o")t

color(red)(ulbar(|stackrel(" ")(" "t = 0.24color(white)(l)"s"" ")|)

It thus takes color(red)(0.24color(white)(l)"seconds" to travel 3.0 "m" horizontally.

" "

(iii)

The time(s) when the height is 4.0 "m":

We can now use the equation

ul(Deltay = v_0sinalpha_0t - 1/2g t^2

where

  • Deltay is the change in vertical position (4.0 "m")

  • v_0 = 25 "m/s"

  • alpha_0 = 30^"o"

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

  • g = 9.81 "m/s"^2

Plugging in known values:

4.0color(white)(l)"m" = (25color(white)(l)"m/s")sin(30^"o")t - 1/2(9.81color(white)(l)"m/s"^2)t^2

Using the quadratic formula yields

color(red)(ulbar(|stackrel(" ")(" "t = 0.375color(white)(l)"s"color(white)(l)"and"color(white)(l)t = 2.17color(white)(l)"s"" ")|)

Therefore, the mass has a height of 4.0 "m" when the time is color(red)(0.375color(white)(l)"seconds" and color(red)(2.17color(white)(l)"seconds".