Question

A Schiefer Wurf: Beim schiefen Wurf mit dem Winkel \(\alpha\) gegenüber der horizontalen \(x\)-Achse hat ein Körper die Anfangsgeschwindigkeit \(v=\left(v_{0} \cdot \cos \alpha ; v_{0} \cdot \sin \alpha\right) .\) Bestimmen Sie die Gleichung der Wurfparabel. B Ein Punkt rotiert gleichmäßig in der \(x-y\)-Ebene. In 2 sec. durchläuft er dreimal die Kreisbahn mit dem Radius \(R\). Geben Sie die Parameterdarstellung an. C a) Welche Kurve wird beschrieben durch die Parameterdarstellung $$ \begin{aligned} &x(t)=t \\ &y(t)=t \\ &z(t)=t \end{aligned} $$ b) Auf welche Kurve führt die folgende Parameterdarstellung: \(x(t)=a \cos t\) \(y(t)=b \sin t\)

Short answer

A: Parametric equations for the oblique throw are \(x(t)=v_{0} \cdot \cos \alpha \cdot t\), \(y(t)=v_{0} \cdot \sin \alpha \cdot t - 0.5 \cdot g \cdot t^2\). B: Parametric representation for circular motion in x-y plane is \(x(t) = R \cos(\omega t)\), \(y(t) = R \sin(\omega t)\). C: a) For \(x(t)=y(t)=z(t)=t\), the curve is a straight line in 3D. b) For \(x(t)=a \cos t, y(t)=b \sin t\), the curve is an ellipse with semi-axes \(a\) and \(b\).

Step-by-step solution

Oblique Throw

The parametric representation of the trajectory for a projected body can be obtained from the equations of motion. Using the initial velocity components and the gravitational acceleration (consider it as \(g\)), the representation can be formed as: \(x(t) = v_{0} \cdot \cos \alpha \cdot t\), \(y(t) = v_{0} \cdot \sin \alpha \cdot t - 0.5 \cdot g \cdot t^2\)

Circular Motion

In the circular motion, the point completes three revolutions in 2 seconds. The angular velocity (\(\omega\)) is \(3 \cdot 2\pi / 2\). Therefore, the parametric representation of the motion can be written as: \(x(t) = R \cos(\omega t)\), \(y(t) = R \sin(\omega t)\)

Identification of Curves

a) The parameter representation \(x(t)=t, y(t)=t, z(t)=t\) represents a straight line in 3D space passing through the origin and making equal angles with the axes. b) The parameter representation \(x(t)=a \cos t, y(t)=b \sin t\) is for an ellipse with semi-major axis \(a\) and semi-minor axis \(b\).

Key concepts

Oblique Throw

Understanding projectile motion is essential when dealing with problems involving an oblique throw. This is a type of motion where an object is thrown into the air at an angle, and its path can be described by physics principles. When an object is projected with an initial velocity at an angle \(\alpha\) above the horizontal x-axis, it has two initial velocity components: \(v_0 \cdot \cos \alpha\) in the horizontal direction and \(v_0 \cdot \sin \alpha\) in the vertical direction.

The object's path, known as the trajectory, is influenced by gravity, which pulls it downwards, leading to a characteristic curved path. To analyze this motion, we break it down into two components: horizontal and vertical. Horizontally, the object travels at a constant speed since there are no forces acting in the horizontal direction after the throw. Vertically, the object's speed changes due to the gravitational acceleration (g), causing it to slow down until it reaches the peak of its trajectory, and then accelerate downwards.

The motion under gravity is symmetric, meaning the time taken to rise to the highest point is equal to the time to fall back to the same horizontal level. The shape of this trajectory is a parabola. Students can visualize this by plotting the parametric equations for x and y over time.

Parametric Representation

A parametric representation expresses the coordinates of points as functions of one or more independent parameters, rather than directly in terms of coordinates themselves. This form is especially useful in physics when dealing with the motion of objects, as it separates variables and allows one to track changes over time.

For instance, in projectile motion, the position of the object is described by separate functions for x (horizontal position) and y (vertical position) as functions of time (t). The parametric equations for an oblique throw are: \(x(t) = v_0 \cdot \cos \alpha \cdot t\) and \(y(t) = v_0 \cdot \sin \alpha \cdot t - 0.5 \cdot g \cdot t^2\). These equations account for constant horizontal velocity and accelerated vertical motion under gravity.

Such representations are not limited to two dimensions – they can be expanded into three dimensions, which is essential in more complex physical scenarios, like analyzing the motion of a particle in space.

Circular Motion

When an object moves in a circular path, it is said to be in circular motion. This is another fundamental concept in physics that can also be analyzed using parametric equations.

In our textbook example, a point moves uniformly in a circle in the xy-plane. The point completes three full revolutions in two seconds. This equates to an angular velocity \(\omega = 3 \cdot 2\pi / 2\) radians per second. The parametric equations \(x(t) = R \cos(\omega t)\) and \(y(t) = R \sin(\omega t)\) describe the circular path, where R is the radius of the circle. Here, the angle \(\omega t\) continuously changes as time progresses, causing the point to follow a circular trajectory. Students can visualize circular motion by plotting these parametric equations and observing the x and y values as time increments.

Identification of Curves

In mathematics and physics, identifying curves described by parametric equations is vital to understand the object's motion. For example, the parametric equations \(x(t)=t, y(t)=t, z(t)=t\) represent a straight line in 3D space because each coordinate component of the point increases uniformly with time, implying that the point moves diagonally through space at a constant rate.

On the other hand, the parametric equations \(x(t)=a \cos t\) and \(y(t)=b \sin t\) describe an ellipse with semi-major axis a and semi-minor axis b. This is because the equations map out the x and y positions of a point that follows an elliptical path, where its maximum distance from the center along the x-axis is a and along the y-axis is b. Different values of 'a' and 'b' change the shape of the ellipse, with a circle being a special case when a and b are equal. Recognizing these curves from their parametric forms is a crucial skill for students to develop as they explore more complex systems in physics.