A cannon located at sea level fires a ball with initial speed 82 m / s and initial angle 45^∘ .

step 1 So here we're going to be using equation 425, and we can then say that y equaling x tangent of t

A cannon located at sea level fires a ball with initial...

A cannon located at sea level fires a ball with initial speed 82 $\mathrm{m} / \mathrm{s}$ and initial angle $45^{\circ}$ . The ball lands in the water after traveling a horizontal distance 686 $\mathrm{m}$ . How much greater would the horizontal distance have been had the cannon been 30 $\mathrm{m}$ higher?

Key Concepts

Projectile Motion

Projectile motion is the study of objects that are launched into the air and subject only to the acceleration due to gravity. It involves decomposing the object's motion into horizontal and vertical components, where the horizontal motion occurs at a constant velocity and the vertical motion experiences constant acceleration. This concept helps analyze the trajectory, time of flight, and landing distance of the object.

Kinematic Equations

The kinematic equations are a set of mathematical formulas that describe the motion of objects under constant acceleration. They are used to determine key quantities such as displacement, velocity, and time in both the horizontal and vertical directions. In projectile motion, these equations allow for the calculation of the projectile's trajectory by relating its initial speed, launch angle, and the acceleration due to gravity.

Range of a Projectile

The range of a projectile is the total horizontal distance it travels before returning to its original vertical position (or meeting a different landing level). For an object launched from ground level, a specific formula involving the initial speed, launch angle, and gravity provides the maximum horizontal distance. Understanding how the range is influenced by these factors is crucial when comparing launch conditions at different heights.

Effect of Launch Height

When a projectile is launched from an elevated position, the additional height extends its time of flight, which in turn increases the horizontal distance traveled. The effect of launch height is incorporated by accounting for the extra vertical displacement in the kinematic equations. This concept is essential in analyzing how a projectile's range changes when the launch point is raised above the ground level.

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