To find the difference in speeds of the two objects when they reach the ground, we need to consider the horizontal and vertical motions separately.
First, let's calculate the time it takes for both objects to reach the ground.
For the object launched horizontally, we know the initial horizontal speed is 12.10 m/s, and there is no horizontal acceleration. Hence, the horizontal motion is uniform, and the time taken to reach the ground can be found using the formula:
time = distance / speed
The distance is the horizontal distance from the object launcher to the ground, which we don't know yet. We'll calculate it later.
For the object that is dropped with no initial speed, we can calculate the time taken to reach the ground using the formula:
time = sqrt(2 * height / gravity)
Here, height is the initial vertical distance from the object to the ground, which is given as 77 cm (converted to meters, it is 0.77 m), and gravity is the acceleration due to gravity, which is approximately 9.8 m/sĀ².
Now, let's calculate the horizontal distance traveled by the launched object.
The formula to calculate horizontal distance is:
distance = speed * time
We already know the initial horizontal speed of the launched object is 12.10 m/s, and we can use the time calculated above to find the horizontal distance.
Finally, we can find the difference in speeds of the two objects when they reach the ground by subtracting the speed of the dropped object from the speed of the launched object.
Let's calculate step by step:
Step 1: Calculate the time taken by the dropped object to reach the ground:
time = sqrt(2 * height / gravity) = sqrt(2 * 0.77 / 9.8) ā 0.307 s
Step 2: Calculate the horizontal distance traveled by the launched object:
distance = speed * time = 12.10 * 0.307 ā 3.7197 m
Step 3: Calculate the speed of the launched object when it reaches the ground. Since the horizontal speed remains constant, this is the initial horizontal speed:
speed_launched_object = 12.10 m/s
Step 4: Calculate the speed of the dropped object when it reaches the ground. Since there is no initial horizontal speed and only vertical speed is involved, this is the final vertical speed, which can be found using the formula:
final_vertical_speed = gravity * time = 9.8 * 0.307 ā 3.0096 m/s
Step 5: Calculate the difference in speeds:
difference_in_speeds = speed_launched_object - final_vertical_speed = 12.10 - 3.0096 ā 9.09 m/s
Therefore, the difference in speeds of the two objects when they reach the ground is approximately 9.09 m/s.