Romeo is chucking pebbles gently up to juliet’s window, and he wants the pebbles to hit the window with only a horizontal component of velocity. He is standing at the edge of a rose garden 4.5 m below her window and 5.0 m from the base of the wall (Fig. 3-34).
How fast are the pebbles going when they hit her window?
To determine the speed at which the pebbles hit Juliet's window, we can use kinematic equations.
First, we need to determine the time it takes for the pebbles to travel from Romeo's position to Juliet's window. We can use the vertical displacement and the acceleration due to gravity to find this time.
Given:
Vertical displacement (h) = 4.5 m
Acceleration due to gravity (g) = 9.8 m/s^2
We can use the following kinematic equation:
h = (1/2) * g * t^2
Substituting the values, we have:
4.5 = (1/2) * 9.8 * t^2
Simplifying the equation:
9 = 4.9 * t^2
t^2 = 9/4.9
Taking the square root:
t ≈ 1.91 s
Now that we have the time taken, we can calculate the horizontal distance traveled by the pebbles using the given distance (5.0 m). We can use the equation:
distance = velocity * time
Rearranging the equation:
velocity = distance / time
Substituting the values, we have:
velocity = 5.0 m / 1.91 s
Calculating the velocity:
velocity ≈ 2.62 m/s
Therefore, the pebbles are going with a horizontal component velocity of approximately 2.62 m/s when they hit Juliet's window.
To determine the horizontal component of velocity, we need to calculate the time it takes for the pebbles to reach the window.
Step 1: Calculate the time of flight:
We can use the equation of motion in the vertical direction to find the time it takes:
y = (1/2)gt^2
where y is the vertical displacement (4.5 m), g is the acceleration due to gravity (9.8 m/s^2), and t is the time of flight.
Rearranging the equation:
t^2 = (2y) / g
t = sqrt[(2 * 4.5) / 9.8]
t ≈ 0.95 seconds
Step 2: Calculate the horizontal distance:
Since we know the distance between Romeo and the base of the wall is 5.0 m, and the velocity in the horizontal direction remains constant, the horizontal distance (d) can be calculated using the equation:
d = v * t
where v is the horizontal component of velocity and t is the time of flight.
Since d = 5.0 m, we can rearrange the equation to solve for v:
v = d / t
v = 5.0 / 0.95
v ≈ 5.26 m/s
Step 3: Calculate the total velocity:
The total velocity can be obtained using the Pythagorean theorem:
v_total = sqrt[(v_horizontal^2) + (v_vertical^2)]
Given that the vertical component of velocity is equal to the acceleration due to gravity (9.8 m/s^2), we can calculate the total velocity:
v_total = sqrt[(5.26^2) + (9.8^2)]
v_total ≈ 11.1 m/s
Therefore, when the pebbles hit Juliet's window, they will have a total velocity of approximately 11.1 m/s.
so, if we name the takeoff point as (0,0) we want the vertex of the parabola to be at (5,4.5)
The parabola can thus be written as
y = -0.18(x-5)^2 + 4.5
so, at x=0, y' = 1.8
That means the initial angle is θ=60.9°
Now you should be able to work out the velocity components, and at the top of the arc, only the x component matters.