A 0.9-kilogram mass has just been dropped from the roof of a building. I need to catch it after it has fallen exactly 92 meters. If I weigh 88 kilograms and start running at 5 meters per second as soon as the object is released, how far away can I stand and still catch the object?
I can stand __ m away and still catch the object. (Type an integer or decimal rounded to two decimal places as needed.)
The correct answer and explanation is :
Step 1: Find the Time for the Object to Fall
The object falls freely under gravity, so we use the kinematic equation:
[
d = v_0 t + \frac{1}{2} g t^2
]
where:
- ( d = 92 ) m (distance fallen),
- ( v_0 = 0 ) m/s (initial velocity),
- ( g = 9.8 ) m/s² (acceleration due to gravity),
- ( t ) is the time to fall.
Since ( v_0 = 0 ), the equation simplifies to:
[
92 = \frac{1}{2} (9.8) t^2
]
[
92 = 4.9 t^2
]
[
t^2 = \frac{92}{4.9} = 18.78
]
[
t = \sqrt{18.78} \approx 4.33 \text{ seconds}
]
Step 2: Find How Far You Can Run
You start running at 5 m/s as soon as the object is dropped. The distance you can cover in time ( t = 4.33 ) s is:
[
\text{distance} = \text{speed} \times \text{time}
]
[
d = 5 \times 4.33
]
[
d = 21.65 \text{ meters}
]
Final Answer:
You can stand 21.65 meters away and still catch the object.
Explanation:
The problem involves a freely falling object and a person running towards the landing spot. We calculated the fall time using kinematic equations, then determined how far you could run during that time. The key was recognizing that your motion is constant speed, while the object’s motion is accelerated. By ensuring both reach the same spot at the same time, we determined that the maximum starting distance you can have is 21.65 meters.