Question

Parabolic trajectory Consider the parabolic trajectory $$ x=\left(V_{0} \cos \alpha\right) t, y=\left(V_{0} \sin \alpha\right) t-\frac{1}{2} g t^{2} $$ where \(V_{0}\) is the initial speed, \(\alpha\) is the angle of launch, and \(g\) is the acceleration due to gravity. Consider all times \([0, T]\) for which \(y \geq 0\) a. Find and graph the speed, for \(0 \leq t \leq T.\) b. Find and graph the curvature, for \(0 \leq t \leq T.\) c. At what times (if any) do the speed and curvature have maximum and minimum values?

Short answer

Answer: The minimum value for both speed and curvature occurs at \(t=0\). The maximum value for both speed and curvature occurs at \(t=T\).

Step-by-step solution

Find the derivatives of the position functions with respect to time

Differentiate the given parametric equations for \(x(t)\) and \(y(t)\) with respect to time \(t\). These will be the components of the velocity vector. $$ \frac{dx}{dt} = V_{0}\cos{\alpha}, $$ $$ \frac{dy}{dt} = V_{0}\sin{\alpha} - gt. $$

Calculate the speed

The speed is the magnitude of the velocity vector. So, we square the derivatives obtained in Step 1 and add them, then take the square root. $$ v(t) = \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} = \sqrt{(V_{0}\cos{\alpha})^2 + (V_{0}\sin{\alpha} - gt)^2}. $$

Graphing the speed

Use a graphing tool to plot \(v(t)\) for \(0 \leq t \leq T\). b) Finding and graphing the curvature:

Find the second derivatives of the position functions with respect to time

Differentiate the velocity components we obtained in Step 1 with respect to time. $$ \frac{d^2x}{dt^2} = 0, $$ $$ \frac{d^2y}{dt^2} = -g. $$

Calculate the curvature

Using the obtained first and second derivatives in Step 1 and Step 4, and plug them into the formula for curvature: $$ \kappa(t) = \frac{\left|\frac{d^2x}{dt^2}\frac{dy}{dt} - \frac{d^2y}{dt^2}\frac{dx}{dt}\right|}{\left(\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2\right)^{3/2}}. $$ Calculate the curvature as a function of time by plugging in the derivatives: $$ \kappa(t) = \frac{g\left|V_{0}\cos{\alpha}\right|}{\left((V_{0}\cos{\alpha})^2 + (V_{0}\sin{\alpha} - gt)^2\right)^{3/2}}. $$

Graphing the curvature

Use a graphing tool to plot \(\kappa(t)\) for \(0 \leq t \leq T\). c) Finding the maxima and minima for speed and curvature:

Analyze the speed and curvature functions

For speed, we can see the minimum value is at \(t=0\) and the maximum value is at \(t=T\). For curvature, observe the \(\kappa(t)\) function and notice that the curvature is always non-negative and increases with time. So, the minimum value is at \(t=0\) and the maximum value is at \(t=T\). To verify this, we can differentiate \(\kappa(t)\) with respect to time and check the sign of the derivative.

Key concepts

Initial Speed

In the study of parabolic trajectories, the initial speed, denoted as \( V_0 \), is the velocity at which an object begins its journey. It is a crucial factor in determining how far and high the object will travel.

Initial speed combines with other factors, such as the angle of launch and gravity, to shape the object's path. The formula \( x=\left(V_{0} \cos \alpha\right) t \) illustrates how initial speed influences horizontal motion, projecting horizontally at a factor determined by the angle \( \alpha \).

• Impacts the total range of the projectile.
• Governs the shape and height of the arc.
• Affects the time spent in the air.

To understand the role of initial speed, think of it as the starting power pushing the object along its path. The greater the initial speed, the further and faster the object travels.

Angle of Launch

The angle of launch, represented by \( \alpha \), is the angle at which a projectile is launched with respect to the horizontal. This angle plays a vital role in defining the trajectory of the projectile.

It influences both vertical and horizontal movements, detailed in the equations: \( y=\left(V_{0} \sin \alpha\right)t - \frac{1}{2} gt^2 \) and \( x=\left(V_{0} \cos \alpha\right) t \).

• Determines the height and range of the projectile's flight.
• A smaller angle yields a shorter but quicker path, while a larger angle provides greater height but less range.
• A 45-degree angle often offers the optimal balance for maximizing distance in the absence of air resistance.

Mastering the angle of launch is like adjusting the aim to hit a target with precision. The right angle ensures that the projectile reaches its desired destination efficiently.

Acceleration due to Gravity

The constant denoted as \( g \), or the acceleration due to gravity, determines the downward force acting on the projectile. This unchanging force pulls objects towards the Earth, altering their vertical motion.

In the equation \( y=\left(V_{0} \sin \alpha\right)t - \frac{1}{2} gt^2 \), gravity affects vertical displacement, creating the familiar parabolic shape.

• Standard value on Earth is approximately \( 9.81 \, \text{m/s}^2 \).
• Causes the projectile to eventually come back to the ground.
• Determines the time a projectile spends in the air.

Understanding gravity is essential for predicting when and where a projectile will land. It's the hidden hand that guides objects back to Earth after their brief flight.

Parametric Equations

Parametric equations provide a mathematical framework to describe the path of a projectile. For parabolic motion, they express the horizontal and vertical positions as functions of time: \( x=\left(V_{0} \cos \alpha\right) t \) and \( y=\left(V_{0} \sin alpha\right) t - \frac{1}{2} gt^2 \).

These equations allow us to trace the complete path of an object and understand how its position changes over time. They are fundamental to analyzing objects in motion.

• Helps determine specific points along the trajectory at given times.
• Supports calculations of speed, distance, and height.
• Useful for graphing the motion of projectiles in two dimensions.

Embracing parametric equations is like having a detailed map of an object's journey. With them, we can accurately predict and visualize the entire trajectory of a projectile throughout its flight.