
Speed, Velocity, and Acceleration
Along with displacement, velocity and acceleration round
out the holy trinity of kinematics. As you’ll see, all three are
closely related to one another, and together they offer a pretty
complete understanding of motion. Speed, like distance,
is a scalar quantity that won’t come up too often on SAT II Physics,
but it might trip you up if you don’t know how to distinguish it
from velocity.
Speed and Velocity
As distance is to displacement, so speed is to velocity:
the crucial difference between the two is that speed is a scalar
and velocity is a vector quantity. In everyday conversation, we usually
say speed when we talk about how fast something is moving. However,
in physics, it is often important to determine the direction of
this motion, so you’ll find velocity come up in physics problems
far more frequently than speed.
A common example of speed is the number given
by the speedometer in a car. A speedometer tells us the car’s speed,
not its velocity, because it gives only a number and not a direction. Speed
is a measure of the distance an object travels in a given length
of time:
Velocity is a vector quantity defined as rate
of change of the displacement vector over time:
average velocity =
It is important to remember that the average speed and
the magnitude of the average velocity may not be equivalent.
Instantaneous Speed and Velocity
The two equations given above for speed and velocity discuss
only the average speed and average velocity
over a given time interval. Most often, as with a car’s speedometer,
we are not interested in an average speed or velocity, but in the instantaneous
velocity or speed at a given moment. That is, we don’t want
to know how many meters an object covered in the past ten seconds;
we want to know how fast that object is moving right now.
Instantaneous velocity is not a tricky concept: we simply take the
equation above and assume that is very, very small.
Most problems on SAT II Physics ask about an object’s
instantaneous velocity rather than its average velocity or speed
over a given time frame. Unless a question specifically asks you
about the average velocity or speed over a given time interval,
you can safely assume that it is asking about the instantaneous
velocity at a given moment.
Example

Instantaneous velocity has a magnitude and a direction,
and deals with the velocity at a particular instant in time. All
three of these requirements are met only in B. A is
an example of average velocity, C is an example of
instantaneous speed, and both D and E are
examples of average speed.
Acceleration
Speed and velocity only deal with movement at a constant
rate. When we speed up, slow down, or change direction, we want
to know our acceleration. Acceleration is a vector quantity
that measures the rate of change of the velocity vector with time:
average acceleration =
Applying the Concepts of Speed, Velocity, and Acceleration
With these three definitions under our belt, let’s apply
them to a little story of a zealous high school student called Andrea.
Andrea is due to take SAT II Physics at the ETS building 10 miles
due east from her home. Because she is particularly concerned with
sleeping as much as possible before the test, she practices the
drive the day before so she knows exactly how long it will take
and how early she must get up.
Instantaneous Velocity
After starting her car, she zeros her odometer so that
she can record the exact distance to the test center. Throughout
the drive, Andrea is cautious of her speed, which is measured by
her speedometer. At first she is careful to drive at exactly 30
miles per hour, as advised by the signs along the road. Chuckling
to herself, she notes that her instantaneous velocity—a vector quantity—is
30 miles per hour due east.
Average Acceleration
Along the way, Andrea sees a new speed limit sign of 40
miles per hour, so she accelerates. Noting with her trusty wristwatch
that it takes her two seconds to change from 30 miles per hour due
east to 40 miles per hour due east, Andrea calculates her average
acceleration during this time frame:
average acceleration =
This may seem like an outrageously large number, but in
terms of meters per second squared, the standard units for measuring
acceleration, it comes out to 0.22 m/s^{2}.
Average Velocity: One Way
After reaching the tall, black ETS skyscraper, Andrea
notes that the test center is exactly 10 miles from her home and
that it took her precisely 16 minutes to travel between the two locations.
She does a quick calculation to determine her average velocity during
the trip:
Average Speed and Velocity: Return Journey
Satisfied with her little exercise, Andrea turns the car
around to see if she can beat her 16minute time. Successful, she
arrives home without a speeding ticket in 15 minutes. Andrea calculates
her average speed for the entire journey to ETS and back home:
Is this the same as her average velocity? Andrea reminds
herself that, though her odometer reads 20 miles, her net displacement—and
consequently her average velocity over the entire length of the
trip—is zero. SAT II Physics is not going to get her with any trick
questions like that!
