Physics I: Lesson 1 - Vectors and Scalars

What are unit vectors?

A unit vector is a vector in a particular direction (usually in one dimension) with the magnitude of one. When given a vector and you want to find the unit vector, you divide the components (vertical and horiztonal) by the vector (because vector divided by vector gives the unit vector of 1). Note: unit vectors in different dimensions can be summed up ihat + ihat, jhat+jhat

What does it mean when it is said that acceleration is the rate of a rate?

Acceleration is the rate or the change of velocity, a rate. To calculate the change in velocity, one can take the derivative of velocity, which is the change in displacement over the change in time. Velocity is the rate of displacement over time and acceleration is the rate of the velocity.

When calculating vectors, how can you find the magnitude of the vector's components, horizontal and vertical? Note: trigonometry

By using trigonometry, and knowing the angle to which the vector projects, one can use the angle, as well as the magnitude of the given vector and use sine and cosine. For cases of a right angle (generated by addition of the horizontal and vertical, you can use pythagorean theroem too)

True or false? An object's average acceleration must be in the same direction as the object's change in velocity. If false, provide a counterexample.

True. An object's average acceleration must be in the same direction as the object's change in velocity.

True or false? An object's acceleration must be in the same direction as the direction of motion, i.e. when an object slows down.

False. An object's acceleration, when slowing down will be in the same direction as the object's change in velocity. But the direction of motion will be in the opposite direction because the object is slowing down.

Like instantaneous velocity, what is instantaneous acceleration?

Instantaneous acceleration is the change in velocity at that moment in time. It is the acceleration as the change in time approaches 0.

Does the average velocity have to equal the instantaneous velocities?

No, the instantaneous velocity is a vector quantity at a specific point in time, while the average velocity is the quantity over the change in time. The average velocity may equal to the instantaneous velocity if the velocity of the object does not change but in general, the average does not have to equal the instantaneous. Instantaneous velocity is the velocity as the change in time approaches 0.

Which of the following are examples of a vector and scalar, respectively? A) Distance is a vector and displacement is a scalar. B) Displacement is a vector and distance is a scalar. C) Speed is vector and mass is a scalar. D) Mass is a vector and speed is a scalar.

The answer is B. A vector is a measurement/quantity of magnitude and direction, while a scalar quantity is a measure of magnitude. Displacement is a vector because it describes a magnitude with a direcion "5 meters left". Speed, distance, and mass are scalar quantities because they have not direction.

How can trigonometry be used in vector calculations?

The components of a vector are essentially the horizontal and vertical lines of the vector, which is the hypotenuse of a triangle that forms from the vector and its components. If the angle (or direction) of the vector is known as well as the magnitude of the vector (the length of the hypotenuse), then one can use sine and cosine to find the corresponding components. Furthermore, if the magnitude of the components and vector are known, then the inverse sine, cosine, or tangent can be used to find the direciton or angle of the vector.

What is the difference between displacement and distance? Speed and velocity?

The velocity and displacement are vectors, with magnitude and direction. Speed and distance are quantities with magnitude. For displacement, this value only concerns the initial and final position of the object but not the path traveled, whereas distance, the path traveled is taken into account. Velocity is the change in displacement over time whereas the speed is the change in distance over time.

What is the relationship between displacement, velocity, and acceleration?

These are all vector quantities, a measurement of magnitude with direction. Displacement describes distance travelled in a particular direction (m). Velocity describes the distance travelled over a period of time in that direction (m/s). Acceleration is the change in speed over time in a particular direction or the change in velocity over time (m/s^2)