High School Physics : Understanding Scalar and Vector Quantities

Study concepts, example questions & explanations for High School Physics

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Example Questions

Example Question #1 : Understanding Scalar And Vector Quantities

Which of the following is a scalar quantity?

Possible Answers:

Displacement

Time

All of these are scalar quantities

Acceleration

Force

Correct answer:

Time

Explanation:

The difference between a scalar and a vector is that a vector requires a direction. Scalar quantities have only magnitude; vector quantities have both magnitude and direction. Time is completely separated from direction; it is a scalar. It has only magnitude, no direction.

Force, displacement, and acceleration all occur with a designated direction.

Important distinctions to know:

Speed is a scalar, while velocity is a vector.

Distance is a scalar, while displacement is a vector.

Force and acceleration are vectors. Time is a scalar.

Example Question #2 : Understanding Scalar And Vector Quantities

Which of the following is a vector quantity?

Possible Answers:

Displacement

All of these are vector quantities

Speed

Distance

Time

Correct answer:

Displacement

Explanation:

A vector has both magnitude and direction, while a scalar has only magnitude. Ask yourself, "for which of these things is there a direction?" For displacement, we would say "50 meters NORTH," whereas with the others, we would say "50 meters," "20 seconds," or "30 miles per hour."

Important distinctions to know:

Speed is a scalar, while velocity is a vector.

Distance is a scalar, while displacement is a vector.

Force and acceleration are vectors. Time is a scalar.

Example Question #21 : Introductory Principles

Michael walks \displaystyle 10m north, \displaystyle 3m west, \displaystyle 5m south, \displaystyle 12m east, and then stops to catch his breath. What is the magnitude of his displacement from his original point?

Possible Answers:

\displaystyle 8.32m

\displaystyle 10.30m

\displaystyle 20.45m

\displaystyle 11.30m

\displaystyle 1.11m

Correct answer:

\displaystyle 10.30m

Explanation:

Displacement is a vector quantity; the direction that Michael travels will be either positive or negative along an axis. We are being asked to solve for his position relative to his starting point, NOT for the distance he has walked.

First we need to find his total distance travelled along the y-axis. Let's say that all of his movement north is positive and south is negative.

\displaystyle \sum Y= 10m-5m=5m. He moved a net of 5 meters to the north along the y-axis.

Now let's do the same for the x-axis, using positive for east and negative for west.

\displaystyle \sum X= -3m+12m=9m. He moved a net of 9 meters to the east.

Now to find the resultant displacement, we use the Pythagorean Theorem. The net movement north will be perpendicular to the net movement east, forming a right triangle. Michael's position relative to his starting point will be the hypotenuse of this triangle.

\displaystyle D^2=(\sum X)^2+(\sum Y)^2

\displaystyle D^2=(9m)^2+(5m)^2

\displaystyle D^2=106m^2

Now take the square root of both sides.

\displaystyle \sqrt{D^2}=\sqrt{106m^2}

\displaystyle D=10.30m

Since the problem only asks for the magnitude of the displacement, we do not need to provide the direction.

Example Question #2 : Understanding Scalar And Vector Quantities

Leslie walks \displaystyle 20m north, \displaystyle 35m east, \displaystyle 10m north, and then \displaystyle 6m west before stopping. What is her displacement from her original location?

Possible Answers:

\displaystyle 21.73m \text{ at }46.3^{\circ}

\displaystyle 33.73m \text{ at }49.3^{\circ}

\displaystyle 41.73m \text{ at }44.03^{\circ}

\displaystyle 41.73m \text{ at }22.3^{\circ}

\displaystyle 53.4m \text{ at }42.3^{\circ}

Correct answer:

\displaystyle 41.73m \text{ at }44.03^{\circ}

Explanation:

Displacement is a vector quantity; it will have both magnitude and direction.

First we need to find his total distance travelled along the y-axis. Let's say that all of her movement north is positive and south is negative.

\displaystyle \sum Y= 20m+10m=30m. She moved a net of 30 meters to the north. 

Now let's do the same for the x-axis, using positive for east and negative for west.

\displaystyle \sum X=35m-6m=29m. She moved a net of 29 meters to the east.

Now, to find the resultant displacement, we use the Pythagorean Theorem. Her net movement north will be perpendicular to her net movement east, forming a right triangle. Her location relative to her starting point will be the hypotenuse of the triangle.

\displaystyle D^2=(\sum X)^2+(\sum Y)^2

\displaystyle D^2=(29m)^2+(30m)^2

\displaystyle D^2=841m^2+900m^2

\displaystyle D^2=1741m^2

Now take the square root of both sides.

\displaystyle \sqrt{D^2}=\sqrt{1741m^2}

\displaystyle D=41.73m

Since we are solving for a vector, we also need to find the direction of this distance. We do this by solving for the angle of displacement.

To find the angle, we use the arctan of our directional displacements in the x- and y-axes. The tangent of the angle will be equal to the x-displacement over the y-displacement.

\displaystyle \tan(\Theta)=\frac{\sum X}{\sum Y}

\displaystyle \Theta =\tan^{-1}({\frac{\sum X}{\sum Y}})

\displaystyle \Theta=44.03^{\circ}

Combining the magnitude and direction of our distance gives us the displacement: \displaystyle 41.73m \text{ at }44.3^{\circ}.

Example Question #3 : Understanding Scalar And Vector Quantities

Angie runs around a circular track for \displaystyle 20mins. The track is \displaystyle 1km and she runs at a rate of \displaystyle \frac{ 1km}{10min}. What is her total displacement?

Possible Answers:

\displaystyle 2km\ clockwise

\displaystyle 0km

\displaystyle 0.2km

\displaystyle 2km\ counterclockwise

\displaystyle 2km

Correct answer:

\displaystyle 0km

Explanation:

Since she is running on a circular track, every time she makes a loop she has a total displacement of \displaystyle 0km. Remember, displacement take into account how far you've travelled, it only uses the total change in distance from where you start and where you stop. Using dimensional analysis, we can determine how many laps she runs in 20 minutes.

\displaystyle 20min*\frac{1km}{10min}*\frac{1lap}{1km}=2\ laps

After twenty minutes, she has made EXACTLY two loops around the track. That means she is starting and stopping in EXACTLY the same place. Her displacement would be \displaystyle 0km, since there is no change between her starting position and her ending position.

Example Question #4 : Understanding Scalar And Vector Quantities

Walter is washing windows on a large building. He starts by washing the window on the 4th floor, then down to the 3rd floor, then up to the 6th floor, then down to the 5th floor, then down to the 2nd floor, and finally he washes the 1st floor window. What is his total distance?

Possible Answers:

\displaystyle 5 \text{ floors}

\displaystyle 9 \text{ floors}

\displaystyle 11 \text{ floors}

\displaystyle 3 \text{ floors}

\displaystyle 13 \text{ floors}

Correct answer:

\displaystyle 9 \text{ floors}

Explanation:

Distance is a scalar quantity and will take into account only the number of floors travelled, regardless of the direction of movement.

Walter takes an incredibly complicated path to wash the windows on the building. When calculating distance, we add up all the movement he does, regardless of direction.

First, he travels down one floor (4th to 3rd).

\displaystyle D_{total}=(1)

Then he travels up three floors (3rd to 6th).

\displaystyle D_{total}=(1)+(3)

Then he travels down one floor (6th to 5th), then down another three floors (5th to 2nd).

\displaystyle D_{total}=(1)+(3)+(1)+(3)

Finally, he travels down one more floor (2nd to 1st).

\displaystyle D_{total}=(1)+(3)+(1)+(3)+(1)=9

In total, Walter travelled \displaystyle 9 \text{ floors}.

Example Question #31 : Introductory Principles

Walter is washing windows on a large building. He starts by washing the window on the 4th floor. He then moves down to the 3rd floor, then up to the 6th floor, then down to the 5th floor, then down to the 2nd floor, and finally he washes the 1st floor window. What is his total displacement?

Possible Answers:

\displaystyle 3\ \text{floors downward}

\displaystyle 9\ \text{floors}

\displaystyle 5 \text{ floors}

\displaystyle 9\ \text{floors downward}

\displaystyle 3\ \text{floors upward}

Correct answer:

\displaystyle 3\ \text{floors downward}

Explanation:

Displacement is a vector relating the starting position to the ending position. Displacement does not take into account the route to arrive at the endpoint, and has both magnitude and direction.

In spite of taking a very complicated route to get there, Walter starts at the 4th floor and ends at the 1st floor.

\displaystyle \vec D=floor_{final}-floor_{initial}

\displaystyle \vec D=1-4

\displaystyle \vec D=-3\ floors

Sine the result is negative, the displacement is 3 floors downward.

Example Question #1 : Understanding Scalar And Vector Quantities

Ariel walks \displaystyle 10m to the east, \displaystyle 7m to the west, and then \displaystyle 3m to the east again. What is her total displacement?

Possible Answers:

\displaystyle 6m\text{ east}

\displaystyle 0m \text{ east}

\displaystyle 20m\text{ east}

\displaystyle 10m\text{ west}

\displaystyle 10m \text{ east}

Correct answer:

\displaystyle 6m\text{ east}

Explanation:

Displacement is a vector quantity, with both magnitude and direction. Remember, displacement does not take into account the route traveled, only the difference between starting position and ending position.

All movement in this question occurs along the x-axis (east and west. We use positive for east and negative for west, since direction is important to measure displacement.

\displaystyle \vec D=10m-7m+3m=6m

This means that her total displacement was \displaystyle 6m\text{ east}.

Example Question #4 : Understanding Scalar And Vector Quantities

Which of these is a vector quantity?

Possible Answers:

Temperature

Speed

Distance

Acceleration

Time

Correct answer:

Acceleration

Explanation:

Scalar quantities give a magnitude, while vector quantities give a magnitude and a direction. The answer will be a measurement that must act in a given direction.

Distance is a measure of length, regardless of the direction. Displacement is the vector equivalent of distance.

Speed is a measure of rate, regardless of direction. Velocity is the vector equivalent of speed.

Temperature and time do not act in any direction and are purely scalar.

Acceleration must act in a given direction, and is a vector. An acceleration is described by both a magnitude and a direction of action.

Example Question #5 : Understanding Scalar And Vector Quantities

Which of these is a scalar quantity?

Possible Answers:

Force

Momentum

Velocity

Mass

Displacement

Correct answer:

Mass

Explanation:

Scalar quantities give a magnitude, while vector quantities give a magnitude and a direction. The answer will be a measurement that does not change, regardless of the direction of action.

Displacement is a measure of length in a given direction; distance is the scalar version of displacement.

Velocity is a measure of rate in a given direction; speed is the scalar version of velocity.

Force is a derivative of acceleration, and can only act in a given direction. There is no scalar equivalent of force. Similarly, momentum is a derivative of velocity and has no scalar equivalent.

Mass is a measure solely of magnitude, and requires no direction of action. Mass is a scalar quantity.

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