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?
To determine how fast the pebbles are going when they hit the window, we need to use the principles of projectile motion.
Projectile motion refers to the motion of an object that is launched into the air and moves along a curved path under the influence of gravity, with no other forces acting upon it horizontally. In this case, the pebble is thrown from the edge of the rose garden towards the window.
The relevant equations that govern projectile motion are:
1. The horizontal motion equation:
- Distance = Speed * Time
2. The vertical motion equations:
- Final velocity (Vf) = Initial velocity (Vi) + Acceleration * Time
- Distance = Initial velocity * Time + (0.5 * Acceleration * Time^2)
In this scenario, we want to find the speed (Vf) of the pebbles when they hit the window, which corresponds to the final velocity in the vertical direction.
First, let's consider the vertical motion. The pebbles are initially at a height (H) below the window and they move downward due to gravity. The acceleration due to gravity is approximately -9.8 m/s² (negative because it is acting against the upward direction). We can use the second vertical motion equation to find the time it takes for the pebbles to fall from the edge of the rose garden to the window.
Using the equation:
H = Vi * Time + (0.5 * -9.8 * Time^2)
Since the initial vertical velocity (Vi) is 0 (assuming the pebbles are not thrown upwards), the equation simplifies to:
H = 0.5 * -9.8 * Time^2
Rearranging the equation, we get:
Time = sqrt((2 * H) / 9.8)
Plugging in the given value of H = 4.62 m, we can calculate the time it takes for the pebbles to fall.
Next, we need to determine the horizontal distance (D) that the pebbles travel before reaching the window. This distance remains constant throughout the motion.
Now, we have the time it takes for the pebbles to fall and the horizontal distance they travel. The final step is to calculate the speed of the pebbles when they hit the window. We can use the horizontal motion equation to determine the speed.
Using the equation:
Speed = D / Time
Plugging in the given values of D = 5.21 m and the previously calculated time, we can find the speed of the pebbles when they hit the window.