Pellets of mass 2.0 g are fired in parallel paths with speeds of 150 m/s through a hole 3.0 mm in diameter. How far from the hole must you be to detect a 1.0-cm-diameter spread in the beam of pellets?

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Convert all given quantities to SI units for consistency. The mass of the pellets is 2.0 g = 0.002 kg, the speed is 150 m/s, the hole diameter is 3.0 mm = 0.003 m, and the desired spread diameter is 1.0 cm = 0.01 m.

Understand that the spread of the beam is due to the angular divergence caused by the finite size of the hole. The angular spread θ can be approximated as θ ≈ d / L, where d is the diameter of the hole and L is the distance from the hole to the point of observation.

The spread of the beam at a distance L is given by Δx = θ × L. Substituting θ ≈ d / L into this equation, we get Δx = (d / L) × L = d. To achieve a spread of 1.0 cm (0.01 m), we need to solve for L.

Rearrange the equation for L: L = Δx / θ. Since θ ≈ d / L, substitute θ = d / L into the equation to get L = Δx / (d / L). Simplify to find L = (Δx × L) / d.

Solve for L explicitly: L = Δx / d. Substitute the values Δx = 0.01 m and d = 0.003 m into the equation to calculate the distance L.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Projectile Motion

Projectile motion refers to the motion of an object that is launched into the air and is subject to gravitational forces. In this scenario, the pellets are fired horizontally, and their trajectory will be influenced by gravity, causing them to spread out as they travel. Understanding the principles of projectile motion helps in calculating the distance over which the pellets will spread.

Kinematics

Kinematics is the branch of physics that deals with the motion of objects without considering the forces that cause the motion. It involves concepts such as velocity, acceleration, and displacement. In this problem, kinematic equations can be used to determine how far the pellets travel before they spread to a specified diameter, taking into account their initial speed and the time of flight.

Beam Divergence

Beam divergence refers to the spreading of a beam of particles or waves as it travels through space. In this case, the pellets will diverge from their initial path due to their finite size and the effects of air resistance. Understanding beam divergence is crucial for calculating the distance required to achieve a specific spread in the pellet beam, as it directly relates to the initial conditions and the geometry of the setup.