Question

The position vector \(r\) describes the path of an object moving in the \(x y\) -plane. Sketch a graph of the path and sketch the velocity and acceleration vectors at the given point. $$ \mathbf{r}(t)=t^{2} \mathbf{i}+t \mathbf{j},(4,2) $$

Short answer

The graph of the path will be a parabolic curve. The velocity vector at the point (4,2) is \(<4,1>\) and the acceleration vector at the same point is \(<2,0>\).

Step-by-step solution

Understanding the position vector

The position vector given is \(r(t) = t^2\mathbf{i} + t\mathbf{j}\). This can be interpreted as the x-coordinate being \(t^2\) and y-coordinate being \(t\) at time \(t\). This is a parametric representation of the path.

Sketching graph of the path

Plot the parametric function \(r(t) = t^2\mathbf{i} + t\mathbf{j}\). For instance, for each unit of time \(t\), plot the corresponding x and y coordinate i.e. (\(t^2\), \(t\)). Note that (4,2) would be a point on this graph.

Finding and sketching velocity vector

The velocity vector v(t) is the derivative of the position vector r(t). Therefore, calculate the derivative of \(r(t) = t^2\mathbf{i} + t\mathbf{j}\) to get the velocity function \(v(t) = 2t\mathbf{i} + \mathbf{j}\). The velocity vector at the point (4,2) is \(v(2) = 4\mathbf{i} + \mathbf{j}\). So from the point (4,2), draw the vector \(<4, 1>\). This is your velocity vector.

Finding and sketching acceleration vector

The acceleration vector a(t) is the derivative of the velocity vector v(t). So find the derivative of \(v(t) = 2t\mathbf{i} + \mathbf{j}\) to get the acceleration function \(a(t) = 2\mathbf{i}\). The acceleration vector at the point (4, 2) is \(a(2) = 2\mathbf{i}\). So from the same point, draw a vector parallel to the x axis with a magnitude of 2. This is your acceleration vector.

Key concepts

Position Vector

Understanding the concept of a position vector is crucial for grasping the fundamentals of motion in calculus. The position vector, typically denoted as \(r(t)\), is a function that gives the coordinates of a point relative to the origin of a coordinate system at a specific instance of time. Imagine it as an arrow pointing from the origin to the object’s location in space.

For instance, in our exercise the position vector is given by \(r(t) = t^2\mathbf{i} + t\mathbf{j}\), where \(t\) represents time, and \(\mathbf{i}\) and \(\mathbf{j}\) are the unit vectors in the x and y directions respectively. This means, to find the position at any time \(t\), you’d square the value of \(t\) for the x-coordinate and keep the value of \(t\) for the y-coordinate.

To visualize this, if we plug in the time \(t=2\), we obtain the point \( (4,2) \) which is part of the path that a moving object follows. Sketching the entire path involves plotting several such points for different values of time and connecting them to form a curve that represents the object’s trajectory.

Velocity Vector

The velocity vector is another key element in understanding motion. It represents the rate of change of the position vector with respect to time, indicating how fast and in which direction an object is moving at any given moment. You can think of it as the speedometer mixed with a compass for the moving object.

In the exercise, the velocity vector \(v(t)\) is obtained by differentiating the position vector \(r(t)\) with respect to \(t\), which gives \(v(t) = 2t\mathbf{i} + \mathbf{j}\). The velocity at the specific moment when \(t=2\) is \(v(2) = 4\mathbf{i} + \mathbf{j}\). This tells us that at time \(t=2\), the object is moving in the direction of the vector \( <4, 1> \) at a certain magnitude, which is visually represented by drawing this vector from the point \( (4,2) \) on the path.

To assist with comprehending the concept, imagine an arrow attached to our moving object which lengthens when the object moves faster and shrinks when it slows down. The direction of the arrow adjusts to show the direction in which the object moves instantly.

Acceleration Vector

The acceleration vector plays a pivotal role in the dynamics of an object’s motion. It is defined as the rate at which the velocity vector changes with time. In simpler terms, it tells us how quickly the velocity is changing, whether the object is speeding up, slowing down, or changing direction.

Back to our exercise, the acceleration vector \(a(t)\) is the derivative of the velocity vector, which in our case is a constant \(a(t) = 2\mathbf{i}\). The acceleration is independent of time in this particular scenario, indicating a constant rate of change in velocity solely in the x-direction. At the point \( (4,2) \) for \(t=2\), the acceleration vector is \(a(2) = 2\mathbf{i}\). The graphical representation of this would be a small horizontal arrow pointing in the positive x-direction from the point (4,2), reflecting the constant acceleration.

An easy way to envision this is to think about driving a car on a straight road. The harder you press the gas pedal, the longer the acceleration arrow becomes - but as long as you’re moving straight, the arrow won’t change direction.