Question

Measurements made in 1910 indicate that the common flea is an impressive jumper, given its size.
Assume that a flea’s initial speed is 2.2 m/s, and that it leaps at an angle of 21° with respect to the
horizontal. If the jump lasts 0.16 s, what is the magnitude of the flea’s horizontal displacement?
How high does the flea jump?

Answer 1

To calculate the flea's horizontal displacement and maximum height during its jump, we need to decompose the initial speed into its horizontal and vertical components and use kinematic equations.

### Given data:
- Initial speed (\( v_0 \)) = 2.2 m/s
- Angle of leap (\( \theta \)) = 21°
- Time of jump (\( t \)) = 0.16 s

### Step 1: Calculate horizontal and vertical components of the initial velocity
The horizontal (\( v_{0x} \)) and vertical (\( v_{0y} \)) components of the initial speed can be calculated using the trigonometric functions cosine and sine respectively.

\[
v_{0x} = v_0 \cdot \cos(\theta)
\]
\[
v_{0y} = v_0 \cdot \sin(\theta)
\]

\[
v_{0x} = 2.2 \cdot \cos(21^\circ) \approx 2.2 \cdot 0.9330 \approx 2.050 m/s
\]
\[
v_{0y} = 2.2 \cdot \sin(21^\circ) \approx 2.2 \cdot 0.3584 \approx 0.790 m/s
\]

### Step 2: Calculate the horizontal displacement
The horizontal displacement (\( x \)) can be computed as:

\[
x = v_{0x} \cdot t
\]
\[
x = 2.050 \cdot 0.16 \approx 0.328 m
\]

### Step 3: Calculate the maximum height
For maximum height, we can use the vertical motion. The maximum height can be calculated using:

\[
h = v_{0y} \cdot t - \frac{1}{2} g t^2
\]

Where \( g \) is the acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \)).

However, since the flea jumps at an angle, we need to first calculate the time to reach the highest point. The time to reach the maximum height can be found using:

\[
t_{up} = \frac{v_{0y}}{g}
\]

Calculating the time to reach maximum height:

\[
t_{up} = \frac{0.790}{9.81} \approx 0.0804 s
\]

Now, we can plug this back into our height equation to find the maximum height:

\[
h = v_{0y} \cdot t_{up} - \frac{1}{2} g t_{up}^2
\]
\[
h = 0.790 \cdot 0.0804 - \frac{1}{2} \cdot 9.81 \cdot (0.0804)^2
\]
Calculating the height:

\[
h = 0.0636 - \frac{1}{2} \cdot 9.81 \cdot 0.00646 \approx 0.0636 - 0.0317 \approx 0.0319 m
\]

### Summary of Results
1. **Horizontal Displacement**: \( \approx 0.328 \, \text{m} \)
2. **Maximum Height**: \( \approx 0.0319 \, \text{m} \) (or about 3.19 cm)

Thus, the flea's horizontal displacement is approximately 0.328 meters, and it jumps to a height of approximately 0.0319 meters (or 3.19 centimeters).