Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. If an object moves on a trajectory with constant speed \(S\) over a time interval \(a \leq t \leq b,\) then the length of the trajectory is \(S(b-a)\) b. The curves defined by \(\mathbf{r}(t)=\langle f(t), g(t)\rangle\) and \(\mathbf{R}(t)=\langle g(t), f(t)\rangle\) have the same length over the interval \([a, b]\) c. The curve \(\mathbf{r}(t)=\langle f(t), g(t)\rangle,\) for \(0 \leq a \leq t \leq b,\) and the curve \(\mathbf{R}(t)=\left\langle f\left(t^{2}\right), g\left(t^{2}\right)\right\rangle,\) for \(\sqrt{a} \leq t \leq \sqrt{b},\) have the same length. d. The curve \(\mathbf{r}(t)=\left\langle t, t^{2}, 3 t^{2}\right\rangle,\) for \(1 \leq t \leq 4,\) is parameterized by arc length.

Short answer

Question: Determine the truth of the given statements regarding curves and provide explanations or counterexamples. a. True: An object moving on a trajectory with constant speed will have a trajectory length equal to the speed multiplied by the time interval. b. True: Curves with vector functions \(\mathbf{r}(t) = \langle f(t), g(t) \rangle\) and \(\mathbf{R}(t) = \langle g(t), f(t) \rangle\) have the same length. c. False: The curve with vector functions \(\mathbf{r}(t) = \langle f(t), g(t) \rangle\) and \(\mathbf{R}(t) = \langle f(t^2), g(t^2) \rangle\) do not necessarily have the same length. A counterexample is when curve \(\mathbf{r}(t)\) is a straight line \(y=t\) from \(t=0\) to \(t=1\), then curve \(\mathbf{R}(t)=\langle t^{2},t^{4} \rangle\) has a different length. d. False: The curve \(\mathbf{r}(t) = \langle t, t^2, 3t^2 \rangle\) is not parameterized by arc length, as the magnitude of its derivative is not equal to one for all values of \(t\).

Step-by-step solution

Understand the definition of speed

Since the object moves with constant speed \(S\), the distance covered in a given time interval \((b-a)\) will always be the same. Speed is the rate of change of distance, so let's use that to find the length of the trajectory.

Calculate the length of the trajectory

The trajectory length can be calculated by multiplying the constant speed with the time interval, \(S(b-a)\). Therefore, the statement is true. b. True or False: The curves defined by \(\mathbf{r}(t) = \langle f(t), g(t) \rangle\) and \(\mathbf{R}(t) = \langle g(t), f(t) \rangle\) have the same length?

Calculate the length of the curves

The length of a curve can be computed using the arc length formula. Since the only difference between the two curves is the order of \(f(t)\) and \(g(t)\), the length of the curves will be the same. Therefore, the statement is true. c. True or False: The curve \(\mathbf{r}(t)=\langle f(t), g(t)\rangle\) and the curve \(\mathbf{R}(t)=\left\langle f\left(t^{2}\right), g\left(t^{2}\right)\right\rangle\) have the same length?

Calculate the length of the curves

As before, we'll use the arc length formula to compute the length of these curves. However, in this case, the curves are not just differing in order, but also their parameterizations. In general, they won't have the same length. Consider a counterexample, when curve \(\mathbf{r}(t)\) is a straight line \(y=t\) from \(t=0\) to \(t=1\), then curve \(\mathbf{R}(t)=\langle t^{2},t^{4} \rangle\) has a different length. Therefore, the statement is false. d. True or False: The curve \(\mathbf{r}(t)=\langle t, t^{2}, 3t^{2}\rangle\) is parameterized by arc length?

Verify if the curve is parameterized by arc length

To parameterize a curve by arc length, we need to check if the magnitude of its derivative is equal to one. Let's find the derivative and its magnitude for the given curve. \(\mathbf{r}'(t) = \langle 1, 2t, 6t \rangle\), \(|\mathbf{r}'(t)| = \sqrt{1^2 + (2t)^2 + (6t)^2} = \sqrt{1 + 4t^2 + 36t^2} = \sqrt{1 + 40t^2}\) Since the magnitude of the derivative is not equal to one for all \(t\) in the given interval, the statement is false. The curve is not parameterized by arc length.

Key concepts

Constant Speed

When we talk about an object moving at a constant speed, we mean that it covers equal distances in equal intervals of time. This is a straightforward concept where the speed, denoted as \(S\), remains unchanged no matter how long or short the time interval is. For instance, if a car is traveling at a constant speed of 60 km/h, it will cover 60 km every hour regardless of the conditions.
In the context of a trajectory, which is a path that an object takes as it moves in space, constant speed simply means that the rate at which the object moves along the path is uniform. Therefore, the length of the trajectory covered over a time interval \([a, b]\) can be calculated using the formula \(S(b-a)\), where \(b-a\) is the elapsed time during which the object maintains its speed. Thus, knowing the constant speed and the time interval, one can easily compute the path length.

Trajectory

A trajectory is essentially the path that an object follows through space as a function of time. This path can be straight or curved, simple or complex, depending on the forces acting on the object. Trajectories are central to understanding motion in physics and engineering because they visually and mathematically describe where an object has been and where it is going.
For example, when a ball is thrown, its trajectory is the curved path it follows while airborne. This path results from the initial force applied to the ball and other forces acting on it, such as gravity and air resistance. Analyzing a trajectory involves studying how these forces influence the object's path and speed over time.
Understanding trajectories allows for predictions about future movement and is essential for tasks such as guiding spacecraft, designing roller coasters, or even animating characters in video games.

Curve Parameterization

Curve parameterization is a method of defining how a point moves along a curve over time. It involves using a parameter, typically denoted as \(t\), to express the position of a point in the form of vector functions. For a 2D curve, parameterization would look like \(\mathbf{r}(t) = \langle f(t), g(t) \rangle\), which means at any parameter \(t\), the point on the curve is located at \(x = f(t)\) and \(y = g(t)\).
Parameters can vary, meaning two curves having different parameterizations might look the same physically but are described differently mathematically. This can change the properties of the curve, such as how quickly an object moves along it. It's also possible for different parameterizations of the same curve to not have the same length, as their expressions define the speed of traversal differently.
Parameterizing curves is essential in fields like computer graphics and robotics, where precise control over motion through space is necessary.

Arc Length Formula

The arc length formula is used to determine the distance along a curve. It's a way of calculating how long the path is when traversed from one point to another. For a smooth curve, you can compute the arc length by integrating the magnitude of the derivative of the curve's parameterization.
For a two-dimensional curve parameterized as \(\mathbf{r}(t) = \langle f(t), g(t) \rangle\), the arc length \(L\) from \(t = a\) to \(t = b\) is given by the integral:\[ L = \int_a^b \sqrt{(f'(t))^2 + (g'(t))^2} \, dt \]
This formula calculates the length by adding up infinitesimal distances along the curve, dealing effectively with the varying direction and rate of change of the parameterization.
The arc length is fundamental in various mathematical and engineering disciplines, such as determining the length of roads or paths for construction, analyzing the curvature of objects for design, or in any scenario where the total travel distance is necessary.