Understanding the Physics Behind an Arrow's Flight: From Work-Energy to Hooke's Law

The core principle of understanding the mechanics of a bow and arrow involves the work-energy principle. This principle is essential for analyzing the motion of projectiles and helps us determine the speed at which an arrow leaves the bow. Let's explore the physics step-by-step with a detailed analysis based on the provided problem.

The Problem and Initial Analysis

A 0.075 kg arrow is fired horizontally by a bow. The bowstring exerts an average force of 65 N over a distance of 0.90 m. We need to calculate the speed of the arrow upon departure from the bow.

Step 1: Calculating the Work Done on the Arrow

The work done by the bowstring on the arrow is given by the formula:

W F × d

where:

F 65 N (force exerted by the bowstring) d 0.90 m (distance over which the force is applied)

Substituting the values:

W 65 N × 0.90 m 58.5 J

Step 2: Relating Work to Kinetic Energy

The work done on the arrow is equal to its change in kinetic energy. The kinetic energy (KE) of the arrow is given by:

KE 0.5 × m × v2

where:

m 0.075 kg (mass of the arrow) v final speed of the arrow

Step 3: Solving for the Speed of the Arrow

Setting the work done equal to the kinetic energy:

58.5 J 0.5 × 0.075 kg × v2

Rearranging to solve for v2:

v2 (2 × 58.5 J) / 0.075 kg 1560 m2/s2

Now, taking the square root to find v:

v √1560 m/s ≈ 39.5 m/s

Alternative Approach: Hooke's Law and Energy Storage

A different but equally valid approach involves considering the elastic nature of the bowstring. The force exerted by the bowstring can be expressed as:

F k × e

where:

F 65 N (force exerted by the bowstring) e 0.90 m (extension or elongation of the bowstring) k 72.2 N/m (spring constant)

Energy stored in the bowstring is:

E 0.5 × k × e2 0.5 × 72.2 N/m × (0.90 m)2 29.4 J

This energy is converted to kinetic energy:

KE 0.5 × m × v2 29.4 J

Solving for v2:

v2 (2 × 29.4 J) / 0.075 kg 780 m2/s2

Thus, the speed of the arrow:

v √780 m/s ≈ 27.93 m/s

Discussion and Further Considerations

The above calculations provide a simplified but useful model of how an arrow is launched. However, in reality, the physics of a bow and arrow are more complex:

Hooke's Law and Bow Stiffness

The force exerted by the bowstring follows Hooke's Law, which states that the force is proportional to the extension. This means the stiffness of the bow and the bowstring play a crucial role in determining the initial speed of the arrow.

Complex Geometry and Moment of Inertia

The geometry of the bow and the moment of inertia of the limbs can affect the acceleration and trajectory of the arrow. These factors make the true physics even more nuanced.

The Arrow Paradox

The release of the arrow presents a fascinating paradox. High-speed photography reveals that the arrow flexes outward and inward as it leaves the bow, allowing it to clear the grip. This oscillatory motion is a testament to the engineering and design of the arrow and the bow.

Over thousands of years, humans have refined these artifacts through trial and error, developing techniques that enable precise and consistent arrow flight. The bow and arrow stand as a remarkable example of human ingenuity and the application of fundamental physics principles.