Question

a body is thrown upward with a velocity of 20m/s. it returns to the ground at the same position after 8 s. sketch a graph

Answer 1

Sure! To analyze the motion of a body thrown upward and returning to the same position, it's helpful to understand the basic principles of kinematics. Here's the detailed sketching process:

1. **Key Parameters:**
- Initial velocity \( v_0 = 20 \, \text{m/s} \)
- Total time (up and down) \( t = 8 \, \text{s} \)
- Gravitational acceleration \( g \approx 9.8 \, \text{m/s}^2 \)

2. **Time to Reach the Peak:**
The body will take half the total time to reach the highest point and the other half to return to the ground.
\[
t_\text{up} = t_\text{down} = \frac{8 \, \text{s}}{2} = 4 \, \text{s}
\]

3. **Maximum Height:**
The maximum height can be determined using the kinematic equation \( v = v_0 - gt \):
\[
0 = 20 \, \text{m/s} - 9.8 \, \text{m/s}^2 \cdot 4 \, \text{s}
\]
Since \( v = 0 \) at the peak, we can also use another kinematic equation to find the maximum height \( h \):
\[
h = v_0 t_\text{up} - \frac{1}{2} g t_\text{up}^2
\]
\[
h = 20 \, \text{m/s} \cdot 4 \, \text{s} - \frac{1}{2} \cdot 9.8 \, \text{m/s}^2 \cdot (4 \, \text{s})^2
\]
\[
h = 80 \, \text{m} - \frac{1}{2} \cdot 9.8 \cdot 16
\]
\[
h = 80 \, \text{m} - 78.4 \, \text{m}
\]
\[
h = 1.6 \, \text{m}
\]

4. **Sketching the Graph:**

- **Position-Time Graph:**

\[
\begin{equation*}
\text{Y-axis: height (m)}
\text{X-axis: time (s)}
\end{equation*}
\]

Draw a parabolic curve that starts at the origin (0,0), reaches a maximum height of 1.6 meters at 4 seconds, and returns back to the height of 0 meters at 8 seconds.

- **Velocity-Time Graph:**

\[
\begin{equation*}
\text{Y-axis: velocity (m/s)}
\text{X-axis: time (s)}
\end{equation*}
\]

Draw a straight line that starts at 20 m/s (initial velocity), descends linearly to 0 m/s at 4 seconds (when the object is at the peak), then continues to -20 m/s at 8 seconds (just before hitting the ground).

**Position-Time Graph:**
```
Height (m)
|
|
1.6 - *
| *
| *
| *
| *
0|__________*________________
0 4 8 Time (s)
```

**Velocity-Time Graph:**
```
Velocity (m/s)
20|-*
|
|
|
|
|
0|________*________________
0 4 8 Time (s)
-20|
```

In these graphs:
- The position-time graph is a parabola, showing the upward and downward motion of the body.
- The velocity-time graph is a straight line, showing the consistent acceleration due to gravity.

These are simplified representations to help visualize the motion of the object.