Problem 14 - University Physics with Modern Physics

Question

The froghopper, \(Philaenus\) \(spumarius\), holds the world record for insect jumps. When leaping at an angle of 58.0\(^\circ\) above the horizontal, some of the tiny critters have reached a maximum height of 58.7 cm above the level ground. (See \(Nature\), Vol. 424, July 31, 2003, p. 509.) (a) What was the takeoff speed for such a leap? (b) What horizontal distance did the froghopper cover for this world-record leap?

Step-by-Step Solution

Answer

Takeoff speed: 3.94 m/s; Horizontal distance: 1.33 m.

Step 1: Identify Given Values

We are given the leap angle \( \theta = 58.0^\circ \) and the maximum height \( h = 58.7 \text{ cm} = 0.587 \text{ m} \). We need to find the takeoff speed \( v_0 \) and the horizontal distance covered.

Step 2: Calculation of Takeoff Speed

Using the formula for maximum height in projectile motion, \( h = \frac{v_{0y}^2}{2g} \), where \( v_{0y} = v_0 \sin \theta \) and \( g = 9.81 \text{ m/s}^2 \). Rearrange to solve for \( v_0 \):\[v_0 = \sqrt{\frac{2gh}{\sin^2 \theta}}.\]Substitute \( g = 9.81 \text{ m/s}^2 \), \( h = 0.587 \text{ m} \), and \( \theta = 58.0^\circ \):\[v_0 = \sqrt{\frac{2 \times 9.81 \times 0.587}{\sin^2(58.0^\circ)}} \approx 3.94 \text{ m/s}.\]

Step 3: Determine Horizontal Component of Velocity

The horizontal component of the takeoff speed is \( v_{0x} = v_0 \cos \theta \). Using \( v_0 = 3.94 \text{ m/s} \) and \( \theta = 58.0^\circ \):\[v_{0x} = 3.94 \cos(58.0^\circ) \approx 2.08 \text{ m/s}.\]

Step 4: Find Time of Flight

The time to reach maximum height is given by \( t_{up} = \frac{v_{0y}}{g} \), where \( v_{0y} = v_0 \sin \theta \). Therefore,\[t_{up} = \frac{3.94 \sin(58.0^\circ)}{9.81} \approx 0.32 \text{ s}.\]Since the flight is symmetric, the total time of flight \( T = 2 \times t_{up} \approx 0.64 \text{ s}.\)

Step 5: Calculate Horizontal Distance Covered

The horizontal distance, or range, \( x \) is given by \( x = v_{0x} \times T \). Substitute \( v_{0x} = 2.08 \text{ m/s} \) and \( T = 0.64 \text{ s} \):\[x = 2.08 \times 0.64 \approx 1.33 \text{ m}.\]

Key Concepts

Kinematics

Kinematics is a branch of mechanics that deals with the motion of objects. It does not concern itself with the forces that cause this motion. In this exercise, we explore a specific case of motion, known as projectile motion. The froghopper's leap is a fascinating example of how we can use kinematics to describe complex movements. We divide the motion into vertical and horizontal components, allowing us to study their behavior individually.

• The vertical motion is influenced by gravity, and we use the maximum height formula to understand it. This is where the froghopper's takeoff speed becomes crucial, as it tells us how fast it must launch to reach a certain height.
• The horizontal motion remains constant during the leap, unaffected by gravity. Here, we use kinematics to determine how far the froghopper travels horizontally. By analyzing both components, we get a comprehensive picture of the leap's dynamics.

Understanding kinematics is essential in physics, as it provides the tools to analyze motion, not only for small insects but also for larger objects in the real world.

Physics Problems

Physics problems often involve scenarios where various physical quantities interact. They teach us to apply theoretical concepts to practical situations. In solving the froghopper's leap, we first identify given values like angle and height. This helps set up the problem effectively.

• Next, we utilize equations of motion to calculate desired quantities, such as the takeoff speed and horizontal distance. These calculations require a careful understanding of how different concepts like angles, velocities, and time of flight interrelate.
• Following a structured approach helps in breaking down complex problems into manageable steps. By treating each step methodically, you can derive the desired results without rushing or making errors.

Physics problems enhance problem-solving skills and help develop logical thinking, both of which are crucial in scientific studies and practical applications.

Two-dimensional Motion

Two-dimensional motion occurs when an object moves in a plane, having both a horizontal and a vertical component. It is crucial when studying projectiles, like the froghopper's record leap.

• This type of motion involves separating the overall movement into independent vertical and horizontal motions. Each component is analyzed separately, using its own set of equations related to time, velocity, and displacement.
• For vertical motion governed by gravity, we use the sine component of the angle to determine initial vertical velocity, helping us find the maximum height. In contrast, the horizontal component relies on the cosine to ascertain how far the froghopper will travel from its starting point.

Understanding two-dimensional motion helps us predict and calculate the movement paths of various objects, from tiny insects to large-segment movements in mechanics. It's an essential concept in physics that broadens our comprehension of how objects interact with their environment.