MCAT Physics Lesson 1: Vectors

*CRB* Rusev started on an xy-plane at (-0.5, 3). He walks so that his final position is (-0.5, -3). What is Rusev's displacement? (A) -6 units (B) 6 units (C) -3 units (D) 3 units

(A) -6 units Displacement = Final Position - Initial Position = Net Distance (in some direction) X-displacement = -0.5 - (-0.5) = 0 Y-displacement = (-3) - 3 = -6 Total displacement is 6 units due South.

*CRB* What would the area underneath a Velocity vs Time graph represent? (A) Displacement (B) Distance (C) Acceleration (D) Time Elapsed

(A) Displacement

If Max got in his car and traveled northwest, with a 63.2° angle between his direction and due west, how far west (in miles) would he have gone if he went 26.6 miles in the northwest direction? (A) 4.6 (B) 12.0 (C) 16.8 (D) 34.2

(B) 12.0 cosθ = adjacent/hypotenuse cos63.2 = adjacent/26.6 adjacent = (26.6)(cos63.2) adjacent = (26.6)(approx. .5 (actual: .451)) adjacent = (approx. 13.3 (actual: .12.0))

Joshua drove 17.45 miles in 23.4 minutes. His total displacement, however, was only 9.44 miles in the north direction. What was his average velocity (in mph)? (A) 13.5 (B) 24.2 (C) 43.6 (D) 67.2

(B) 24.2 Average velocity = displacement/change in time Average Velocity = (9.44 miles) / (23.4/60 hour) Average Velocity = (9.44 miles) / (approx. 1/3 hour) Average Velocity = approx. 30 mph (actual: 24.2)

Johnny was traveling at a speed of 14.2 m/s and traveled a total of 345.9 meters. How long did it take? (A) 13.78 s (B) 24.36 s (C) 31.92 s (D) 55.24 s

(B) 24.36 s Average velocity = displacement/change in time 14.2 = 345.9 / change in time change in time = 345.9 / 14.2 change in time = approx. 23 seconds (actual: 24.36)

*CRB* In which of the following examples could you calculate the missing variable from given information? I. Given Distance and Initial Position, find Final Position. II. Given Initial Position and Distance, find Final Position. III. Given Initial Position and Final Position, Find Displacement. (A) I only (B) III only (C) I and II only (D) I, II and III

(B) III only If you know the Initial and Final position, you can find Displacement. However, knowing distance cannot help to find Initial or Final Positions, since distance has no direction, only magnitude!

Scalar is to ____________ as vector is to___________. (A) distance, distance (B) distance, displacement (C) displacement, displacement (D) displacement, distance

(B) distance, displacement Displacement includes a direction and a quantity; therefore, it is a vector quantity. Distance, on the other hand, is a scalar quantity as it only entails a quantity.

*CRB* What would the slope of a Velocity vs Time graph represent? (A) Displacement (B) Distance (C) Acceleration (D) Time Elapsed

(C) Acceleration

Ali is running at 1.22 m/s in the positive direction. Over 19.44 seconds he increases his speed to 2.31 m/s, what is his average acceleration? (A) 3.1 m/s^2 (B) 0.52 m/s^2 (C) 0.31 m/s^2 (D) 0.056 m/s^2

(D) 0.056 m/s^2 Acceleration = change in velocity / change in time Acceleration = ( 2.31 - 1.22 m/s) / (19.44 s) Acceleration = 0.05 m/s^2 (actual: .056 m/s^2)

Unit vectors have a magnitude of:

1

How is acceleration calculated from velocity?

Acceleration = change in velocity/change in time

How is average velocity calculated from displacement?

Average velocity = displacement/change in time

If we know the angle of a vector and adjacent horizontal vector, how can we use that to find the length of the original vector (the hypotenuse)?

By using cosθ = adjacent/hypotenuse

*CRB* In order to use the above equation, you need to know the displacement. How do you calculate displacement?

Displacement = Final Position - Initial Position = Net Distance (in some direction) Shortest distance between two points

*CRB* True or false? If an object's velocity is zero, then the object's acceleration must also be zero. If false, provide a counterexample.

False. Even if an object's velocity is zero, the object could still be accelerating. Think of a baseball thrown straight up into the air; there is a split second where the ball is not moving up or down (at it peak height), but gravity is still making the ball accelerate back towards the Earth.

Match the following xyz-plane directions with their corresponding unit vectors. I. x II. y III. z (A)h^ (B)î (C)ĵ (D)k^

I (B) x: î II (C) y: ĵ III (D) z: k^

What does "instantaneous" mean when referring to velocity (or speed of acceleration, etc)?

It means the velocity at a specific time, not an average over a period of time.

What does soh cah toa stand for?

Sinθ = opposite/hypotenuse Cosθ = adjacent/hypotenuse Tanθ = opposite/adjacent

What does the slope at a point on a position vs time graph represent?

The instantaneous velocity at that point.

*CRB* 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.

*CRB* True or false? Displacement, unlike Distance, can be used when knowing one of Final or Initial to find the other because Displacement has both a magnitude and a direction.

True. Displacement, unlike Distance, can be used when knowing one of Final or Initial to find the other because Displacement has both a magnitude and a direction.

*CRB* True or false? When multiplying a vector by a positive (n>0) number, the magnitude of the vector will change, but not its direction.

True. When multiplying a vector by a positive (n>0) number, the magnitude of the vector will change, but not its direction.

With vectors when broken down into its components (right triangle), how can you find the length of the original vector without the use of angles?

Use the pythagorean theorem

What does vector b̂ = 4î + 3ĵ mean?

Vector b can be broken down into a horizontal vector of length 4 and a vertical vector of length 3.

What is the difference between a vector and a scalar?

Vectors have magnitude (size) AND direction. Scalars just have magnitude (size). Ex. An object moved 5m is a scalar wheras an object moved 5m to the right is a vector

What is the difference between velocity and speed?

Velocity is a vector quantity, while speed is a scalar quantity. Speed = absolute value of velocity.

Say for example you have: b̂ = 4î + 3ĵ a=2î + 4ĵ how would you add these two vectors, and support with tail to tip method

c= 6î + 7ĵ

What would a negative value for acceleration represent?

deceleration or slowing down

A person starts off at 0 and moves 10 m to the right and then 5 m to the left, what is distance vs displacement?

distance: 15m displacement : 5 m Note distance is total area traveled and displacement is final position- initial position Also note, if the person came back to the left 10 m then displacement would've been 0