A shell is fired from a cannon at an angle of α=45 with an initial speed of v0=800 m / s (a) Fin

step 1 For the problem here, we have a shell that's fired from a cannon, and we're going to want to fin

A shell is fired from a cannon at an angle of α=45 with an...

A shell is fired from a cannon at an angle of $\alpha=45$ with an initial speed of $v_{0}=800 \mathrm{m} / \mathrm{s}$ (a) Find parametric equations for the shell's trajectory relative to the coordinate system in Figure $\mathrm{Ex}-36$ (b) How high does the shell rise? (c) How far does the shell travel horizontally?

A shell is fired from a cannon at an angle of $\alpha=45$ with an initial speed of $v_{0}=800 \mathrm{m} / \mathrm{s}$
(a) Find parametric equations for the shell's trajectory relative to the coordinate system in Figure $\mathrm{Ex}-36$
(b) How high does the shell rise?
(c) How far does the shell travel horizontally?

Key Concepts

Projectile Motion

Projectile motion refers to the motion of an object that is launched into the air and is subject only to the acceleration due to gravity. This type of motion follows a curved, parabolic trajectory and can be analyzed by breaking it into horizontal and vertical components that are dealt with independently, except that time is common to both motions.

Parametric Equations

Parametric equations express the horizontal and vertical positions of an object as functions of time, providing a convenient way to represent the trajectory of a projectile. In projectile motion, the horizontal position is often given by a linear function while the vertical position is given by a quadratic function due to the influence of gravity.

Initial Velocity Components

When analyzing projectile motion, the initial velocity is typically decomposed into horizontal and vertical components using trigonometric functions based on the launch angle. This decomposition allows one to separately apply constant velocity motion in the horizontal direction and accelerated motion in the vertical direction, which is crucial for solving problems related to projectile trajectories.

Maximum Height

The maximum height reached by a projectile is determined by the vertical component of its initial velocity and the acceleration due to gravity. By setting the vertical velocity equal to zero at the peak of the trajectory, one can derive the formula for maximum height, illustrating how the initial conditions and gravity interact to limit the projectile’s ascent.

Range Determination

The range of a projectile is the horizontal distance it covers during its flight before returning to the same vertical level from which it was launched. It depends on both the horizontal component of the initial velocity and the total time of flight, which is influenced by the vertical motion under gravity. The range is maximized when the projectile is launched at a specific angle (usually 45° in ideal conditions without air resistance).

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