To find the velocity of impact of the object on the ground, we can break the problem down into two components: the horizontal motion and the vertical motion. The object is thrown horizontally, and we can analyze how it behaves as it falls from the height of 3 meters while also moving horizontally with the bus.
Understanding the Parameters
Let's clarify the quantities given in the problem:
- Horizontal velocity of the object (Vx): 6 m/s
- Velocity of the bus (which the object has initially): 36 km/h (convert this to m/s)
- Height of the window from the ground (h): 3 m
Unit Conversion
First, we need to convert the bus’s velocity from kilometers per hour to meters per second:
1 km/h is equal to 1/3.6 m/s. Therefore:
36 km/h = 36 / 3.6 = 10 m/s
Since the object is thrown horizontally, its initial horizontal velocity (Vx) will be the same as the bus's velocity plus the velocity it was thrown with:
Initial horizontal velocity (Vx) = 10 m/s + 6 m/s = 16 m/s
Vertical Motion Analysis
Next, let’s analyze the vertical motion. The object is falling from a height of 3 meters. To find out how long it takes to hit the ground, we can use the following kinematic equation:
h = (1/2)gt²
Where:
- h = height (3 m)
- g = acceleration due to gravity (approximately 9.81 m/s²)
- t = time in seconds
Rearranging the equation to solve for time (t):
t² = (2h) / g
Substituting the values:
t² = (2 * 3 m) / 9.81 m/s² ≈ 0.612 s
Now, taking the square root gives us:
t ≈ 0.78 s
Calculating the Vertical Velocity at Impact
To find the vertical velocity just before impact (Vy), we use the equation:
Vy = gt
Substituting the values we have:
Vy = 9.81 m/s² * 0.78 s ≈ 7.66 m/s
Final Velocity Calculation
The final velocity of the object just before it hits the ground is the combination of its horizontal and vertical components. We can calculate the magnitude of the final velocity (Vf) using the Pythagorean theorem:
Vf = √(Vx² + Vy²)
Substituting the values:
Vf = √((16 m/s)² + (7.66 m/s)²)
Vf = √(256 + 58.56) ≈ √314.56 ≈ 17.7 m/s
Conclusion
Thus, the velocity of impact of the object on the ground is approximately 17.7 m/s. This result combines both the horizontal motion from the bus and the vertical motion due to gravity. This example illustrates how different components of motion interact and can be analyzed separately before combining them for a final result.
