# what is the relationship between the slope of velocity vs.time graph&ave acceleration?

—extended thanks :-p

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The slope of the graph represents the rate at which the velocity is changing, and that is the definition of acceleration. In terms of calculus, the acceleration is the derivative or differential coefficient of the velocity with respect to time.

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

Which of the follow sentences contains an example of instantaneous velocity?

(A) “The car covered 500 kilometers in the first 10 hours of its northward journey.”

(B) “Five seconds into the launch, the rocket was shooting upward at 5000 meters per second.”

(C) “The cheetah can run at 70 miles per hour.”

(D) “Moving at five kilometers per hour, it will take us eight hours to get to the base camp.”

(E) “Roger Bannister was the first person to run one mile in less than four minutes.”

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/s2.

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 16-minute 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!

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