Measurements made in 1910 indicate that the common flea is an impressive jumper, given its size.

Question

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).