An object is dropped from a height H above the ground. This free-falling object requires 0.994 s to travel the last 38 m before hitting the ground. From what height H above the ground did the object fall? The acceleration of gravity is 9.8 m/s 2 . Answer in units of m

We can solve this problem using the equations of motion for a free-falling object under constant acceleration due to gravity.

Given dataWe can solve this problem using the equations of motion for a free-falling object under constant acceleration due to gravity.

Given 
- Acceleration due to gravity, \( g = 9.8 \, \text{m/s}^2 \)
- Time taken to fall the last 38 m, \( t = 0.994 \, \text{s} \)
- Distance for the last part of the fall, \( d = 38 \, \text{m} \)

### Step 1: Calculate the velocity at the start of the last 38 m of the fall

We can use the second equation of motion to find the velocity \( v_0 \) at the start of the last 38 meters:
\[
d = v_0 t + \frac{1}{2} g t^2
\]
Substitute the given values:
\[
38 = v_0 (0.994) + \frac{1}{2} (9.8) (0.994)^2
\]
Simplifying the equation:
\[
38 = 0.994 v_0 + 0.5 \times 9.8 \times (0.994)^2
\]
\[
38 = 0.994 v_0 + 4.849
\]
\[
33.151 = 0.994 v_0
\]
\[
v_0 = \frac{33.151}{0.994} \approx 33.35 \, \text{m/s}
\]
So, the velocity \( v_0 \) at the start of the last 38 meters is approximately \( 33.35 \, \text{m/s} \).

### Step 2: Calculate the total height \( H \)

Now, we use the first equation of motion to find the height \( H \) from which the object was dropped. At the start of the last 38 meters, the object had a velocity of \( v_0 \), and it started from rest at height \( H \). The relation is given by:
\[
v_0^2 = 2 g H
\]
Substituting the values for \( v_0 \) and \( g \):
\[
(33.35)^2 = 2 \times 9.8 \times H
\]
\[
1112.12 = 19.6 H
\]
\[
H = \frac{1112.12}{19.6} \approx 56.74 \, \text{m}
\]

Therefore, the object was dropped from a height \( H \approx 56.74 \, \text{m} \).

Would you like more details on the steps or have any questions?

Here are 5 related questions for further practice:
1. What is the velocity of the object just before it hits the ground?
2. How would the time change if the object fell from twice the height?
3. What would be the time taken to fall from the full height H?
4. If the acceleration due to gravity were different, how would that affect the result?
5. Can this method be applied to objects with air resistance?

**Tip:** When dealing with free-fall problems, always identify known variables (distance, time, velocity) and choose the appropriate equation of motion.

An object is dropped from a height H above the ground. This free-falling object requires 0.994 s to travel the last 38 m before hitting the ground. From what height H above the ground did the object fall? The acceleration of gravity is 9.8 m/s². Answer in units of m.

Learn how to calculate the height from which an object is dropped using free fall equations. This problem involves the final 38 meters of the fall with a time of 0.994 seconds and gravity acceleration of 9.8 m/s². Suitable for high school students (Grades 10-12).

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