• MC_UNLV
In summary, the conversation discusses finding the magnitude of velocity and acceleration of a rocket, given its distance from a radar station and the angle it makes with the ground. The solution involves finding r(dot) and using the relationship between r and theta as functions of time. The unit vectors r and theta are used in this calculation, with the ^ representing the unit vector. Differentiating these expressions allows for finding r double dot, and understanding polar coordinates is necessary for this problem.
MC_UNLV
A rocket is fired vertically and tracked by a radar station on the ground, a distance (r) away from the rocket. When the station reads an angle of (theta) = 60* between the rocket and the ground, we are given that the distance r = 30,000ft, r(double-dot) = 70 ft/sec, and theta(dot) = 0.02 rad/sec. Find the magnitude of the velocity and acceleration of the rocket at this position.

I know that to solve this, you need to find r(dot), and that this is somehow related to r as a function of time. I do not understand how to get this relationship, or how to find r(dot). Can anyone please help?

Let

$$\vec{r}=r(\theta{})\hat{r}$$

Differentiating with respect to time and using the chain rule gives

$$\vec{\dot{r}}=\dot{r}\hat{r}+r\frac{d\hat{r}}{d\theta{}}\frac{d\theta{}}{dt}$$

and

$$\frac{d\hat{r}}{d\theta{}}=\hat{\theta}$$

Why? Differentiate this expression again to arrive at an expression for r double dot in terms of the unit vectors r and theta. This should get you started.

Forgive my ignorance, but what does the "^" above r and theta mean, and what is the difference between the r with and without the ^?

The ^ represents the unit vector. In Cartesian coordinates it's

$$\hat{x}\mbox{ and }\hat{y}$$

The r without the hat (^) is the magnitude of r. Have you been exposed to polar coordinates and the associated unit vectors?

Yes, I understand vectors, I just have seen it with different notations.

I still do not understand what you are trying to say with the expressions in your first reply. I do not get how to relate time to the values of r and theta, if a specified time is not given.

## 1. What is polar kinematics?

Polar kinematics is a branch of mechanics that studies the motion of objects in a coordinate system where position and velocity are described using polar coordinates, such as distance and angle.

## 2. How is polar kinematics different from Cartesian kinematics?

In Cartesian kinematics, position and velocity are described using Cartesian coordinates (x and y), while in polar kinematics, they are described using polar coordinates (distance and angle).

## 3. What are some common applications of polar kinematics?

Polar kinematics is commonly used in the study of rotational motion, such as the motion of planets and satellites in orbit, as well as in engineering applications involving circular or rotational motion, such as gears and pulleys.

## 4. How do you calculate velocity and acceleration in polar kinematics?

Velocity in polar kinematics is calculated by taking the derivative of the polar coordinates with respect to time, while acceleration is calculated by taking the second derivative. These calculations can be done using the equations for polar velocity and acceleration, or by converting the polar coordinates to Cartesian coordinates and using the equations for Cartesian velocity and acceleration.

## 5. What is the difference between polar displacement and polar distance?

Polar displacement refers to the change in position from one point to another, while polar distance refers to the total distance traveled along a curved path. In other words, polar displacement is a vector quantity, while polar distance is a scalar quantity.

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