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 (the least possible speed).
He is standing at the edge of a rose garden at H = 4.62 m below her window and at D = 5.21 m from the base of the wall. How fast are the pebbles going when they hit her window?
(in m/s)
The pebbles will be going at a speed of 8.45 m/s when they hit Juliet's window.
To find the speed of the pebbles when they hit Juliet's window, we can use the principle of projectile motion. The speed of the pebbles will be the horizontal component of the initial velocity.
First, let's break down the given information:
- The vertical distance between Romeo and Juliet's window is H = 4.62 m.
- The horizontal distance between Romeo and the base of the wall is D = 5.21 m.
When Romeo throws the pebbles, they will follow a parabolic path due to gravity. The vertical distance followed by the pebbles is determined by the height difference, H.
To determine the initial vertical velocity, we can use the equation for vertical displacement in projectile motion:
H = (1/2) * g * t^2
Where g is the acceleration due to gravity (9.8 m/s^2) and t is the time taken by the pebble to reach the window.
Rearranging the equation, we get:
t = sqrt((2H) / g)
Now, let's calculate the time taken by the pebbles to reach the window.
t = sqrt((2 * 4.62) / 9.8)
t ≈ 0.96 seconds
Since we are interested in the horizontal component of the velocity, we can use the horizontal distance, D, and the time, t, to find the initial horizontal velocity.
The equation for horizontal velocity is:
Vx = D / t
Vx = 5.21 / 0.96
Vx ≈ 5.4208 m/s
Therefore, the pebbles will hit Juliet's window with a horizontal velocity of approximately 5.4208 m/s.
To determine the speed of the pebbles when they hit Juliet's window, we can use the principles of projectile motion.
First, let's analyze the given information:
- The vertical displacement (H) between Romeo and Juliet's window is 4.62 m.
- The horizontal displacement (D) from the base of the wall to Romeo's position is 5.21 m.
Since Romeo wants the pebbles to hit the window with only a horizontal component of velocity, we can ignore the vertical displacement (H). This means that the pebbles will follow a horizontal trajectory.
The horizontal motion of the pebbles can be described by the equation: Dx = Vx * t
where Dx is the horizontal displacement, Vx is the horizontal component of velocity, and t is the time.
Since the horizontal displacement (Dx) is given as 5.21 m, we can rearrange the equation to solve for Vx:
Vx = Dx / t
To determine the time (t), we can use the equation for time of flight in projectile motion:
t = sqrt(2 * H / g)
where g is the acceleration due to gravity (approximately 9.8 m/s^2).
Substituting the given value of H, we get:
t = sqrt(2 * 4.62 / 9.8)
Calculating the value of t, we find:
t ≈ 0.961 s
Now we can substitute the values of Dx = 5.21 m and t = 0.961 s into the equation for Vx to find the horizontal component of velocity:
Vx = 5.21 / 0.961
Calculating the value of Vx, we get:
Vx ≈ 5.41 m/s
Therefore, the pebbles will be traveling at approximately 5.41 m/s when they hit Juliet's window.