Translational Motion

Question 1

What SI unit and common variables are associated with length?

Answer 1

meters (m)

The variable d and x are used for distance, while h is used for height, z for depth, and r for radius.

Question 2

What SI unit and common variables are associated with time?

Answer 2

seconds (s)

The variable t is used for time; T is used for period and also has SI units of seconds.

Question 3

What SI unit and common variables are associated with area?

Answer 3

meters² (m²)

The variable A is used for area, while S is used for surface area.

Question 4

What SI unit and common variables are associated with volume?

Answer 4

meters³ (m³)

The variable V is used for volume.

Question 5

What SI unit and common variables are associated with velocity?

Answer 5

meters/seconds (m/s)

The variable v is used for velocity (a vector). Note that m/s are also used to measure speed (a scalar).

Question 6

What SI unit and common variables are associated with acceleration?

Answer 6

meters/seconds² (m/s²)

The variable a is used for acceleration (a vector).

Question 7

What two characteristics are necessary to define a vector?

Answer 7

1. magnitude
2. direction

Displacement, velocity, and acceleration are vector quantities.

Question 8

What is the difference between distance and displacement?

Answer 8

Distance is a scalar, while displacement is a vector.

Distance includes only magnitude, not direction; it is equal to the number of total meters traveled. Displacement includes both magnitude and direction and is equal to the number of meters between the starting and ending points.

Question 9

What is the difference between speed and velocity?

Answer 9

Speed is a scalar, while velocity is a vector.

Speed includes only magnitude, not direction; velocity includes both magnitude and direction. Velocity can have a negative sign, while speed is always positive.

Question 10

What is the magnitude of a vector?

Answer 10

It is the quantity, size, or amount of a measurement.

Magnitude alone is a scalar value, since it lacks direction.

Question 11

What is the direction of a vector?

Answer 11

It is a piece of information that provides spacial orientation, angle, or path.

By convention, right and up (two perpendicular directions) are fixed as positive, while their opposites (left and down) are negative.

Question 12

What is the magnitude of a velocity vector that points directly down at 5 m/s?

Answer 12

5 m/s

Magnitude is the quantity, size, or amount.

Question 13

What is the direction of a velocity vector that points directly up at 13 m/s?

Answer 13

directly upwards

Direction is the angle, orientation, or path.

Question 14

Describe the general process for vector addition.

Answer 14

Vectors are added “tip to tail,” forming a continuous pathway from the first arrow to the last one.

The sum, or resultant, vector is made by connecting the original tail to the final tip.

Question 15

Describe the general process for vector subtraction.

Answer 15

Vectors are subtracted in the same process as addition: “tip to tail”. The direction of the vector being subtracted must be reversed prior to adding them.

Question 16

Define:

component vectors

Answer 16

It represent the magnitudes of a given vector along the x axis and y axis.

Generally, these are referred to as the “x component” and the “y component.” In the image below, vector A is shown with its x and y components as A_x and A_y, respectively.

Question 17

Give the formula for the x component of a force in the xy plane.

Answer 17

F_x = F_total cos θ

In general, [vector]_x = [original vector] * cos θ.

Question 18

What is the x component of velocity for a ball thrown upwards at 4 m/s, 30º to the horizon?

Note that sin(30º) = 0.5 and cos(30º) = 0.86.

Answer 18

3.4 m/s

v_x = 4 cos (30º) = 4 (.86) = 3.4 m/s

Question 19

Give the formula for the y component of a force in the xy plane.

Answer 19

F_y = F_total sin θ

In general, y vector = original vector * sin θ.

Question 20

What is the y component of velocity for a ball thrown upwards at 4 m/s, 30º to the horizon?

Note that sin(30º) = 0.5 and cos(30º) = 0.86.

Answer 20

2 m/s

v_y = 4 sin (30º) = 4 (.5) = 2 m/s

Question 21

What formula relates displacement, time, and acceleration?

Answer 21

Δx = v₀t + 1/2 at²

When dealing with an object that starts at rest, v₀ is equal to zero and the first term can be canceled out. When dealing with cases of zero acceleration, as for horizontal velocity with no air resistance, the second term can be canceled out.

Question 22

Define:

instantaneous velocity

Answer 22

It is the direction and magnitude of the rate of change of distance per unit time.

v = Δx / t

Question 23

Give the instantaneous velocity for a 3 kg block with a momentum of 12 kg*m/s.

Answer 23

4 m/s

Momentum is equal to velocity times mass; in variables, p = mv. Here, 12 = (3)(v), so v must be equal to 4 m/s.

Question 24

Define:

average velocity (v_ave)

Answer 24

It is the total displacement and direction traveled from an initial position, divided by the total time.

v_ave = Δx_total / Δt_total

or

v_ave = (v_f + v_i)/2

Note that the second equation assumes constant acceleration, which is also assumed on the MCAT unless specified otherwise.

Question 25

Give the average velocity for a 1 kg block after it falls from rest for 2 seconds.

Answer 25

9.8 m/s

v = (1/2) at²

v_f = (1/2) (9.8) (2)² = 2 (9.8) = 19.6 m/s

v_i = 0 (given as “rest”)

v_ave = (v_f + v_i) / 2

v_ave = (19.6 + 0) / 2 = 9.8 m/s

Question 26

What formula(s) can be used to find final velocity when given initial velocity?

Answer 26

v_f = v_i + at

or v_f² = v_i² + 2aΔx

Where:

v_f = final velocity (m/s)
v_i = initial velocity (m/s)
a = acceleration (m/s²)
Δx = displacement (m)
t = time (s)

Question 27

Give the velocity of a 1 kg block after it falls from rest for 2 seconds.

Answer 27

19.6 m/s

v_f = v_i + at

v_f = 0 + (9.8)(2) = 2(9.8)
19.6 m/s

Question 28

Define:

acceleration

Answer 28

It is the rate at which velocity changes per unit of time.

a = Δv / t or a = Δx / t²

Question 29

What change must have been made to acceleration if the same distance can now be traveled in half the time?

Answer 29

multiplied by 4

Since a is proportional to Δx / t², halving t quadruples a.

Question 30

What is the acceleration that is always associated with an object in free fall?

Answer 30

9.8 m/s²

This is the acceleration due to gravity, commonly called g. On the MCAT, any falling object is assumed to be in free fall (no air resistance) unless told otherwise.

Note that while it’s convenient to use 10 m/s² for calculations, the MCAT expects you to still choose the correct answer mathematically.

Question 31

How can the maximum height of a moving projectile be found?

Answer 31

Finding the point where the vertical component of velocity, v_y, equals zero.

The object moves upwards until it reaches this point, and afterwards is accelerated downwards by gravity. If the projectile arc is symmetric, this position occurs after half of the total time in flight has elapsed.

Question 32

What is the maximum height of a ball thrown directly upwards from the ground with a velocity of 10 m/s?

Answer 32

5m

The kinematics equation that most easily solves this question is:

v_f² = v₀² + 2aΔx

Set v_f equal to 0 m/s to represent the velocity at the maximum height. v₀ = 10 m/s, and a = -10m/s².

0 = 100 + 2(-10)Δx
Δx = 5m

Question 33

Define:

air resistance

Answer 33

It is the frictional force that opposes an object moving in free fall.

F_air = -kv²

The force of air resistance is proportional to the square of the velocity of the object falling. The constant k is a function of the object’s shape and surface area. Note that you do not need to memorize this equation, but be aware of the presence or absence of air resistance in every projectile motion problem.