A stone falls freely from rest for 8.2 s. (Assume that the positive direction is upward.) (a) Calculate the stone's velocity after 8.2 s. (b) What is the stone's displacement during this time?

To solve this problem, we'll use the following kinematic equations for free fall under the influence of gravity:

1. **Velocity after a certain time (v)**:
   \[
   v = u + at
   \]
   Where:
   - \(v\) is the final velocity,
   - \(u = 0\) m/s (since the stone falls from rest),
   - \(a = -9.8\) m/s² (the acceleration due to gravity, negative because gravity acts downward),
   - \(t = 8.2\) s (the time the stone falls).

2. **Displacement during the fall (s)**:
   \[
   s = ut + \frac{1}{2} a t^2
   \]
   Again, \(u = 0\) m/s, \(a = -9.8\) m/s², and \(t = 8.2\) s.

### Part (a): Calculate the stone's velocity after 8.2 s.
We use the first equation:
\[
v = 0 + (-9.8) \times 8.2 = -80.36 \, \text{m/s}
\]
So, the velocity of the stone after 8.2 seconds is \( v = -80.36 \, \text{m/s} \).

### Part (b): What is the stone's displacement during this time?
Now, using the second equation:
\[
s = 0 + \frac{1}{2} (-9.8) (8.2)^2 = \frac{1}{2} \times (-9.8) \times 67.24 = -329.39 \, \text{m}
\]
Thus, the stone's displacement during the fall is \( s = -329.39 \, \text{m} \).

### Final answers:
- (a) Velocity after 8.2 seconds: \( v = -80.36 \, \text{m/s} \)
- (b) Displacement after 8.2 seconds: \( s = -329.39 \, \text{m} \)

Would you like more details or have any questions?

### 5 Related Questions:
1. What would be the displacement if the stone fell for 10 seconds?
2. How would the displacement change if the stone was thrown upwards initially?
3. What would be the stone's velocity if it fell for 5 seconds instead of 8.2 seconds?
4. How long would it take for the stone to reach a velocity of \( -100 \, \text{m/s} \)?
5. If air resistance were considered, how would it affect the stone's velocity and displacement?

### Tip:
In problems involving free fall, always keep track of the signs of velocity and displacement: downward motion is usually negative when assuming upwards as positive.

A stone falls freely from rest for 8.2 s. (Assume that the positive direction is upward.) (a) Calculate the stone's velocity after 8.2 s. (b) What is the stone's displacement during this time?

Learn how to calculate the velocity and displacement of a stone falling freely from rest for 8.2 seconds under the influence of gravity. This problem covers kinematic equations for free fall, including velocity after 8.2 seconds and displacement during the fall. Suitable for students in Grades 9-11.

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