Question

A stone thrown off a bridge at 20 m above a river has an initial velocity of 12 m/s at an angle of 45° above the horizontal. What is the range of the stone? At what velocity does the stone strike the water?

Answer 1

(a) x= 26m

(b) vf= 23.15m/s

Step-by-step explanation:

Given data

h=20m

Ɵ=45°

to find

(a) range of stone=x=?

(b) velocity=vf=?

Solution

For part (a)

You need to solve for time first using

yf = yi + visinƟt + 1/2gt^2

0 = 20m + 12sin45t + 1/2(-9.8)t^2

and use the quadratic equation to solve for t

t = 3.064 sec

To solve for the distance traveled use

x = xi + vicosƟt + 1/2at^2 there is no acceleration in the x direction so that cancels

x = 12cos(45)(3.064)

x= 26m

For part(b)

For b I'm not sure if you what direction you want the final velocity in the x, y, or the direction its traveling so I'll just give all 3.

Theres no change in the velocity in the x direction so its just vfx = vixcosƟ = 12cos45 = 8.49m/s

For the y direction its vfy^2 = viy^2 + 2g(Δy)

vfy = sqrt((12sin(45))^2 + 2(-9.8)(0-20m)) = 21.54m/s

The velocity the direction the stone is traveling is vf = sqrt(vx^2 + vy^2) = sqrt(8.49^2 + 21.54^2)

vf= 23.15m/s

Answer 2

a)
`x=26m`

b)
`v_(y)=-21.5m/s`

Step-by-step explanation:

From the exercise we know initial velocity, initial height

`y_(o)=20m`

`v_(o) =12m/s`
`\beta =45`º

a) The range of the stone is defined by how far does it goes. From the theory of free falling objects, we have:

`y=y_(o)+v_(oy)t+(1)/(2)gt^(2)`

The stone strike the water at y=0

`0=20+12sin(45)t-(1)/(2)(9.8)t^(2)`

Solving for t, using the quadratic formula

`t=-1.33s` or
`t=3.06s`

Since time can't be negative, the answer is t=3.06s

Now, we can calculate the range of the stone

`x=v_(o)t=(12cos(45))m/s(3.06s)=26m`

b) We can calculate the velocity were the stone strike the water using the following formula

`v_(y)=v_(oy)+gt=12sin(45)m/s-(9.8m/s^(2))(3.06s)=-21.5m/s`

The negative sign indicates that the stone is going down