On level ground a shell is fired with an initial velocity of 40.0 m/s at 60.0° above the horizontal and feels no appreciable air resistance. Find its maximum height above the ground.

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Identify the components of the initial velocity. The initial velocity is given as 40.0 m/s at an angle of 60.0° above the horizontal. Use trigonometric functions to find the vertical component (v_y) and the horizontal component (v_x) of the velocity. The vertical component can be found using v_y = v_0 * sin(θ), and the horizontal component using v_x = v_0 * cos(θ), where v_0 is the initial velocity and θ is the angle.

Calculate the vertical component of the initial velocity using the formula: v_y = 40.0 m/s * sin(60.0°). This will give you the initial velocity in the vertical direction, which is crucial for determining the maximum height.

Understand that at the maximum height, the vertical velocity becomes zero. Use the kinematic equation for vertical motion: v_y^2 = v_0y^2 - 2 * g * h_max, where v_y is the final vertical velocity (0 m/s at maximum height), v_0y is the initial vertical velocity, g is the acceleration due to gravity (approximately 9.81 m/s²), and h_max is the maximum height.

Rearrange the kinematic equation to solve for the maximum height: h_max = (v_0y^2) / (2 * g). Substitute the initial vertical velocity calculated in step 2 and the value of g into this equation to find the maximum height.

Ensure all units are consistent and double-check calculations for accuracy. The maximum height is the result of the vertical motion analysis, and it represents the peak of the projectile's trajectory.

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주요 개념

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Projectile Motion

Projectile motion refers to the motion of an object thrown or projected into the air, subject to only the acceleration of gravity. It involves two components: horizontal and vertical motion. Understanding the initial velocity and angle of projection is crucial for analyzing the trajectory and determining key points like maximum height.

Kinematic Equations

Kinematic equations describe the motion of objects without considering the forces that cause the motion. For vertical motion, the equation v^2 = u^2 + 2as can be used to find the maximum height, where v is the final velocity (0 at the peak), u is the initial vertical velocity, a is the acceleration due to gravity, and s is the displacement.

Trigonometric Decomposition

Trigonometric decomposition involves breaking down the initial velocity into horizontal and vertical components using sine and cosine functions. For a projectile launched at an angle θ, the vertical component is calculated as v_y = v * sin(θ), which is essential for determining the maximum height reached by the projectile.