In the absence of air resistance, a projectile that lands at the elevation from which it was launched achieves maximum range when launched at a 45° angle. Suppose a projectile of mass m is launched with speed into a headwind that exerts a constant, horizontal retarding force Fwᵢₙd = -Fwᵢₙd î. b. By what percentage is the maximum range of a 0.50 kg ball reduced if Fwᵢₙd = 0.60 N?

Verified step by step guidance

1. Identify the initial conditions and parameters given in the problem: mass of the projectile m = 0.50 kg, initial speed v₀ (not specified but necessary for calculations), launch angle θ = 45°, and the horizontal retarding force due to wind Fwᵢₙd = 0.60 N.
2. Use the equation of motion in the horizontal direction considering the retarding force. The net force in the horizontal direction is F_net = -Fwᵢₙd, leading to the acceleration a_x = F_net / m. Calculate the horizontal acceleration a_x using a_x = -Fwᵢₙd / m.
3. Determine the time of flight t of the projectile. Since there is no vertical acceleration other than gravity, use the vertical motion equation y = v₀y * t - 0.5 * g * t², where v₀y is the initial vertical component of the velocity v₀ * sin(θ) and g is the acceleration due to gravity. Solve for t when y = 0 (return to the same elevation).
4. Calculate the horizontal range R of the projectile using the horizontal motion equation x = v₀x * t + 0.5 * a_x * t², where v₀x is the initial horizontal component of the velocity v₀ * cos(θ).
5. Compare the calculated range R with the range R₀ that would be achieved without the wind (Fwᵢₙd = 0) to find the percentage reduction in range. Use the formula for percentage reduction: Reduction (%) = [(R₀ - R) / R₀] * 100.

Key Concepts

Projectile Motion

Projectile motion refers to the motion of an object that is launched into the air and is subject to gravitational force. The trajectory of a projectile is typically parabolic, and its range is influenced by the launch angle and initial velocity. In ideal conditions, the maximum range occurs at a 45° angle, balancing vertical and horizontal components of motion.

Effect of Air Resistance

Air resistance, or drag, is a force that opposes the motion of an object through the air. In this scenario, a headwind exerts a constant horizontal force that reduces the projectile's horizontal velocity, thereby decreasing its range. Understanding how drag affects projectile motion is crucial for calculating the new range when external forces are present.

Force and Acceleration

According to Newton's second law, the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. In this case, the retarding force due to the headwind affects the net force acting on the projectile, leading to a change in its acceleration and ultimately reducing its range. This relationship is essential for quantifying the impact of the wind on the projectile's motion.