A student decides to measure the muzzle velocity of the...

A student decides to measure the muzzle velocity of the pellets from his BB gun. He points the gun horizontally. On a vertical wall a distance $x$ away from the gun, a target is placed. The shots hit the target a vertical distance $y$ below the gun. (a) Show that the vertical displacement component of the pellets when traveling through the air is given by $y=A x^{2}$, where $A$ is a constant. (b) $\mathrm{Ex}$ press the constant $A$ in terms of the initial velocity and the free-fall acceleration. (c) If $x=8.00 \mathrm{~m}$ and $\gamma=$ $0.210 \mathrm{~m}$, what is the initial speed of the pellets?

Key Concepts

Projectile Motion

This is the study of objects that are thrown or projected into the air and then move under the influence of gravity. In projectile motion, the object follows a curved path called a trajectory, which is determined by the initial velocity and the constant acceleration due to gravity acting only in the vertical direction.

Separation of Motion Components

In projectile motion, the horizontal and vertical components of the motion are analyzed independently. The horizontal motion is at constant velocity (assuming no air resistance), while the vertical motion is subject to constant acceleration due to gravity. This separation allows the use of different kinematic equations for each direction and simplifies the analysis of the trajectory.

Kinematic Equations for Uniform Acceleration

These equations describe the motion of objects under constant acceleration, such as gravity. They relate displacement, velocity, time, and acceleration. In projectile motion, these equations are used to calculate the vertical displacement and to relate the time of flight to the horizontal distance traveled, thereby linking the projectile’s motion to its initial velocity.

Free-Fall Acceleration

This is the constant acceleration experienced by an object when falling under the influence of gravity alone, typically denoted by g (approximately 9.81 m/s² near the Earth's surface). It plays a key role in determining how fast an object accelerates downwards in projectile motion, affecting the vertical displacement over time.

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