Problem 84 - Algebra and Trigonometry

Question

Use a graphing utility to obtain the path of a projectile launched from the ground \((h=0)\) at the specified values of \(\theta\) and \(v_{0} .\) In each exercise, use the graph to determine the maximum height and the time at which the projectile reaches its maximum height. Also use the graph to determine the range of the projectile and the time it hits the ground. Round all answers to the nearest tenth. \(\theta=35^{\circ}, v_{0}=300\) feet per second

Step-by-Step Solution

Answer

The maximum height, the time it reaches the maximum height, the range of the projectile, and the time it hits the ground, could be determined from the graph of the projectile. Values would depend on the specific output of the graphing utility. Please use one for exact values, this step-by-step procedure just provides a way of how to use the utility.

Step 1: Identify Given Variables

Identify the given values. Here, \(\theta = 35^{\circ}\) which is the angle of projection, and \(v_{0} = 300\) feet/second which is the initial velocity of the projectile. The acceleration due to gravity \(g\) can be taken as \(32.2\) feet/second\(^2\).

Step 2: Projectile Motion Formulae

The height \(h\) and distance \(x\) as a function of time \(t\) can be given by the equations \(h(t) = v_{0} t \sin(\theta) - 0.5 g t^{2}\) and \(x(t) = v_{0} t \cos(\theta)\). Here, \(g\) is the acceleration due to gravity, \(v_{0}\) is the initial velocity, \(\theta\) is the angle of projection and \(t\) is the time. Convert the angle \(\theta\) from degrees to radians since the trigonometric functions in the calculator use radians. The conversion is done using the relation \(1^{\circ} = \frac{\pi}{180}\) radians.

Step 3: Obtain The Path Of The Projectile

Use a graphing utility to plot the above equations for \(x(t)\) and \(h(t)\). The resulting plot is called the trajectory of the projectile, representing how it moves through space.

Step 4: Determine Maximum Height and Time

The peak point in the graph represents the maximum height that the projectile reaches. Read the coordinates of this point from the graph. The \(y\)-coordinate represents the maximum height, and the \(x\)-coordinate represents the time at which this height is reached.

Step 5: Determine The Range and Time Of Flight

The x-coordinate where the trajectory graph meets the x-axis represents the range of the projectile, which is the total distance in the horizontal direction that the projectile has travelled. The corresponding y-coordinate is the time at which the projectile hits the ground.

Key Concepts

Angle of Projection

The angle of projection is crucial in determining the path a projectile will take. When you launch a projectile, the angle at which you release it, known as the angle of projection, greatly influences its path or trajectory.
For instance, in our exercise, the angle of projection is given as 35 degrees. As a fundamental rule:

• A small angle results in a sharp, low trajectory.
• A 45-degree angle typically offers the maximum range.
• A larger angle forms a higher, shorter path.

In trigonometry, angles are usually considered in radians. To convert degrees to radians, use the formula:\[\theta (\text{radians}) = \theta (\text{degrees}) \times \frac{\pi}{180}\]By applying this conversion, you ensure that calculations based on trigonometric functions remain consistent and accurate. Always remember that the angle of projection is just as important as the velocity in determining how far and high your projectile will go.

Initial Velocity

The initial velocity dictates both the speed and the direction of the projectile at the moment of launch. In this case, the initial velocity is 300 feet per second.
It's a vector quantity, meaning it has both magnitude (speed) and direction (dictated by the angle of projection).To understand its effect, break the initial velocity into two components:

• Horizontal velocity: \(v_{0} \cos(\theta)\)
• Vertical velocity: \(v_{0} \sin(\theta)\)

These components help in predicting the projectile's behavior: - The horizontal component influences how far the projectile travels. - The vertical component, combined with gravity, determines how high it goes and the duration it stays in the air.The greater the initial velocity, the further and higher a projectile can go, assuming the angle of projection remains constant. As you analyze projectile motion, remember that both the speed at launch and the chosen angle significantly shape the resulting trajectory.

Gravitational Acceleration

Gravitational acceleration is the downward force exerted by the Earth's gravity on the projectile. In our exercise, this value is standardized as 32.2 feet per second squared. It acts continuously downward, impacting the projectile's vertical motion.In equations, gravitational acceleration \(g\) appears as:\[h(t) = v_{0} t \sin(\theta) - 0.5 g t^{2}\]The term \(-0.5 g t^{2}\) depicts the constant acceleration by gravity, whereby:

• It decreases the upward speed of the projectile until it reaches the peak.
• Once the projectile starts descending, gravity increases its speed downward.

Understanding gravitational acceleration is important for predicting the motion of a projectile. It consistently pulls the projectile towards the ground, impacting the maximum height and ensuring the projectile returns to land.

Trajectory Analysis

Trajectory analysis involves studying the path taken by a projectile. In our exercise, this path arises from the interplay of the angle of projection, initial velocity, and gravitational acceleration.
The trajectory is typically parabolic, forming a curved arch, representing both horizontal and vertical displacements over time.In this exercise, plotting the function: - \(x(t) = v_{0} t \cos(\theta)\) - \(h(t) = v_{0} t \sin(\theta) - 0.5 g t^{2}\)helps visualize the projectile's path.Key points to identify from the graph include:

• Maximum Height: Achieved when the vertical velocity component is zero, just before it descends.
• Time to Maximum Height: Found as the x-coordinate at the highest point on the trajectory graph.
• Range: The horizontal distance covered, observed at where the trajectory meets the baseline (ground).
• Total Time of Flight: The time taken from launch to when the projectile hits the ground.

Trajectory analysis gives insight into a projectile’s performance, allowing for precise predictions of its flight path, height, and distance travelled.