Velocity, Speed, Acceleration, and Deceleration

The goal for today is to better understand what we mean by terms such as velocity, speed, acceleration, and deceleration. Let’s start with an example, namely the motion of a ball thrown upward and then acted upon by gravity. A major source of confusion in problems of this sort has to do with blurring the distinction between speed and velocity. The speed s is, by definition, the magnitude of the velocity vector: s := |v|. The change in velocity is uniformly downward.

The speed is decreasing during the upward trajectory, and increasing during the subsequent downward trajectory. The laws of physics are most simply written in terms of velocity, not speed. Physics uses a technical definition of acceleration that conflicts with ordinary vernacular use of the words “acceleration” and “deceleration”. That’s tough. You’ll have to get used to it if you want to do physics. In physics, acceleration refers to a change in velocity, not speed. If you want to be really explicit, you can call this the vector acceleration.

In the vernacular, “acceleration” commonly means speeding up, i. e. an increase in speed. If you insist on using the word in this sense, you can remove the ambiguity by calling it the scalar acceleration.

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The scalar acceleration can be considered one component of the vector acceleration, namely the projection in the “forward” direction (although this is undefined if the object is at rest).

In physics, the word “deceleration” is not much used. In particular, it is not the opposite of acceleration, or the negative of acceleration.

Any change in velocity is called an acceleration.  In the vernacular, “deceleration” commonly means slowing down, i. e. a decrease in speed.

Do not confuse the vector acceleration with the scalar acceleration. In physics, acceleration does not mean speeding up. To repeat: In physics, the term acceleration is defined to be the change in velocity, per unit time. It is a vector. This term applies no matter how the acceleration is oriented relative to the initial velocity. There are several possible orientations.

This is a classic example of a situation where the scalar acceleration is zero even though the vector acceleration is nonzero. | Acceleration at some odd angle relative to the velocity. No good way to describe it in terms of scalars. | Acceleration of an object at a moment when its velocity is zero. No way to describe it in terms of scalars; the scalar acceleration formula produces bogus expressions of the form 0/0.

1. To decrease the velocity of.
2. To slow down the rate of advancement

Problem #1:  A skater goes from a standstill to a speed of 6. 7 m/s in 12 seconds. What is the acceleration of the skater?

Step 1: Write down the equation needed for solving for acceleration.

Step 2: Insert the known measurements into the equation.

Known: The initial speed of the skater was zero since he was not in motion. The skater finally reached a speed of 6. 7m/s in 12 seconds, which is the final speed or velocity.

Step 3: Solve. Carefully put all measurements into your calculator.

You must solve the change in velocity portion of the equation before you can do the division portion to solve for acceleration. Don't forget that the SI unit for acceleration is m/s2  .

Practice Problems

1. As a shuttle bus comes to a normal stop, it slows from 9. 00m/s to 0. 00m/s in 5. 00s. Find the average acceleration of the bus.
2. During a race, a sprinter increases from 5. 0 m/s to 7. 5 m/s over a period of 1. 25s.What is the sprinter’s average acceleration during this period?
3. A baby sitter pushing a stroller starts from rest and accelerates at a rate of  0. 500m/s2. What is the velocity of the stroller after it has traveled for 4. 75 minutes?

• A bicyclist accelerates at 0. 89ms2 during a 5. 0s interval. What is the change in the speed of the bicyclist and the bicycle?
• A freight train traveling with a speed of 18. 0m/s begins braking as it approaches a train yard. The train’s acceleration while braking is -0. 33m/s2. What is the train’s speed after 23 seconds?
• A skater travels at a constant velocity of 4. m/s westward, then speeds up with a steady acceleration of 2. 3m/s2. Calculate the skater’s speed after accelerating for 5. 0s.

Solving for Time

• Marisa’s car accelerates at an average rate of 2. 6m/s2. Calculate how long it takes her car to accelerate from 24. 6m/s to 26. 8m/s.
• If a rocket undergoes a constant total acceleration of 6. 25m/s2, so that its speed increases from rest to about 750m/s, how long will it take for the rocket to reach 750m/s.
• A dog runs with an initial speed of 1. 5m/s on a waxed floor. It slides to a stop with an acceleration of -0. 5m/s2. How long does it take for the dog to come to a stop?

Additional acceleration problems (with answers)

1. A body with an initial velocity of 8 m/s moves with a constant acceleration and travels 640 m in 40 seconds. Find its acceleration.
2. A box slides down an inclined plane with a uniform acceleration and attains a velocity of 27 m/s in 3 seconds from rest. Find the final velocity and distance moved in 6 seconds (initially at rest).
3. A car has a uniformly accelerated motion of 5 m/s2. Find the speed acquired and distance traveled in 4 seconds from rest.

4. A marble is dropped from a bridge and strikes the water in 5 seconds. Calculate the speed with which it strikes and the height of the bridge. 

5. A ship starts at rest and reaches a speed of 83 km/h. Suppose it took 2. 0 minutes for the ship to reach that speed. What is the acceleration of the ship?

Answers

1. A body with an initial velocity of 8 m/s moves with a constant acceleration and travels 640 m in 40 seconds. Find its acceleration. (a = 0. 4 m/s2) 

2. A box slides down an inclined plane with a uniform acceleration and attains a velocity of 27 m/s in 3 seconds from rest.

   Find the final velocity and distance moved in 6 seconds (initially at rest). (Vf = 54 m/s, d = 162 m)
3. A car has a uniformly accelerated motion of 5 m/s2. Find the speed acquired and distance traveled in 4 seconds from rest. (Vf = 20 m/s, d = 40m)
4. A marble is dropped from a bridge and strikes the water in 5 seconds. Calculate the speed with which it strikes and the height of the bridge. (Vf = 49 m/s, d = 122 m)
5. A ship starts at rest and reaches a speed of 83 km/h. Suppose it took 2. 0 minutes for the ship to reach that speed. What is the acceleration of the ship?

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