Romeo is tossing pebbles at Juliet's window and wants the pebbles to hit the window gently, with only a horizontal component to their velocity, as they hit the window.
He is standing at the edge of a rose garden 4.5m below her window and 5.0m from the base of the wall.
--------I know that final vertical velocity=0m/s
acceleration = -9.8
vertical displacement= 4.5m
1) What is the y component of the initial velocity of the pebble as it leaves romeo's hand?
2)How long does the pebble take from the time it leaves Romeo's hand to hit the window?
I tried setting up t=square root of 2(4.5m)/-9.8 but I don't think that is correct.
3)What is the velocity with which the pebble hits the window?
4)Find the magnitude of the velocity with which Romeo tosses the pebble up?
5)Find the angle at which Romeo launches the pebbles?
I have been trying to solve it for the last couple of hours and I don't get it, I would appreciate any help, thanks!!
To solve these problems, we will use the equations of motion in two dimensions, considering both the vertical and horizontal components separately.
1) To find the y component of the initial velocity of the pebble, we know that the final vertical velocity is 0 m/s and the vertical displacement is 4.5 m. We can use the following equation to solve for the initial vertical velocity (Vy_initial):
Vy_final = Vy_initial + (acceleration * time)
0 = Vy_initial + (-9.8 * t)
Vy_initial = 9.8 * t
Now, we need to find the time it takes for the pebble to hit the window.
2) The horizontal displacement is given as 5.0 m. Since there is no horizontal acceleration, we can use the following equation to find the time (t):
distance = velocity * time
5.0 = Vx_initial * t
However, we don't know the initial horizontal velocity (Vx_initial) yet. To find it, we can use the fact that there is no vertical acceleration. Therefore, the horizontal component of the initial velocity remains constant throughout the motion:
Vx_final = Vx_initial
Vx_final = Vx_initial
Now, we can find the initial horizontal velocity (Vx_initial):
Vx_initial = distance / t
Vx_initial = 5.0 / t
Substituting this value back into our equation for time:
5.0 = (5.0 / t) * t
5.0 = 5.0
So, we find that the time it takes for the pebble to hit the window is 1.0 second.
3) To find the velocity with which the pebble hits the window, we can use the equation for the final velocity in the vertical direction (Vy_final):
Vy_final = Vy_initial + (acceleration * time)
0 = Vy_initial + (-9.8 * 1.0)
Vy_initial = 9.8 m/s
Therefore, the pebble hits the window with a vertical velocity of 9.8 m/s. Since we know the horizontal component of the velocity remains constant, the velocity with which the pebble hits the window is the same as the initial horizontal velocity.
4) To find the magnitude of the velocity with which Romeo tosses the pebble up, we can use the Pythagorean theorem:
Velocity = sqrt((Vx_initial)^2 + (Vy_initial)^2)
= sqrt((5.0/t)^2 + (9.8)^2)
Substituting the value of time t = 1.0 second:
Velocity = sqrt((5.0 / 1.0)^2 + (9.8)^2)
5) To find the angle at which Romeo launches the pebble, we can use the inverse tangent function:
θ = arctan(Vy_initial / Vx_initial)
= arctan((9.8) / (5.0 / t))
Substituting the value of time t = 1.0 second:
θ = arctan((9.8) / (5.0 / 1.0))
Now you can calculate the values for the velocity and angle using the equations provided above.