As an expert archer you are able to fire off an arrow with a maximum velocity of \(50 \mathrm{~m} / \mathrm{s}\) when you pull the string a length of \(70 \mathrm{~cm}\). (a) If you assume that the acceleration of the arrow is constant from you release the arrow until it leaves the bow, what is the acceleration of the arrow?

In physics, constant acceleration occurs when the rate of change of velocity of an object remains the same over a period of time. This means the object speeds up or slows down by the same amount each second. In the context of the archer's problem, once the arrow is released, it accelerates at a constant rate until it reaches the final velocity when leaving the bow. This simplification makes it easier to use kinematic equations to solve for the unknown values. Constant acceleration is a core assumption in many problems involving motion because it simplifies calculations and provides a good approximation in many real-world scenarios.

Kinematic equations are essential tools in mechanics that describe the motion of objects. These equations relate key variables such as displacement, initial velocity, final velocity, acceleration, and time. For instance, one of the primary kinematic equations used in our archer's problem is: \[v^2 = u^2 + 2ad \] Here:

• \textbf{v} represents the final velocity.
• \textbf{u} is the initial velocity.
• \textbf{a} stands for acceleration.
• \textbf{d} indicates the distance traveled.

Given that the initial velocity (\textbf{u}) is zero when the arrow starts from rest, the equation simplifies to: \[ v^2 = 2ad \] Such equations are powerful for solving motion problems where certain variables are known, and others need to be determined. Understanding and correctly applying these kinematic equations is crucial for solving a wide range of physics problems.

Final velocity refers to the speed of an object at the end of a period of acceleration. In the archer's problem, the final velocity of the arrow (\textbf{v}) is given as 50 meters per second (\textbf{m/s}). This represents how fast the arrow is traveling when it leaves the bow after being accelerated over a certain distance. To find the final velocity, you can use various kinematic equations, depending on what other information is available, such as initial velocity, acceleration, and distance traveled. In this specific scenario, since the initial velocity (\textbf{u}) is zero and the acceleration is constant, the equation used is: \[v^2 = 2ad \] This helps us find unknowns like acceleration or distance if we have the final velocity.

Distance is the measure of how far an object moves during its motion. In the archer's problem, the distance (\textbf{d}) is the length the arrow travels while being accelerated, which is 70 centimeters or 0.70 meters. This value is crucial for determining the acceleration when combined with the final velocity. By using the kinematic equation: \[ v^2 = 2ad \] we clearly see that distance plays a key role in connecting the final velocity and acceleration. It helps form a complete picture of the arrow's motion, allowing for precise calculation of how it reaches its final speed. Understanding the importance of each variable, especially distance, enriches your grasp of motion concepts in physics.