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Motion With Constant Acceleration (Examples, Exercises, Solutions)

An abstraction often used in physics is motion with constant acceleration. This is a good approximation for many different situations: free fall over small distances or in low-density atmospheres, full braking in car traffic, an object sliding down an inclined plane, etc … The mathematics behind this special case is relatively simple. Assume the object that is subject to the constant acceleration a (in m/s²) initially has a velocity v(0) (in m/s). Since the velocity is the integral of the acceleration function, the object’s velocity after time t (in s) is simply:

1) v(t) = v(0) + a · t

For example, if a car initially goes v(0) = 20 m/s and brakes with a constant a = -10 m/s², which is a realistic value for asphalt, its velocity after a time t is:

v(t) = 20 – 10 · t

After t = 1 second, the car’s speed has decreased to v(1) = 20 – 10 · 1 = 10 m/s and after t = 2 seconds the car has come to a halt: v(2) = 20 – 10 · 2 = 0 m/s. As you can see, it’s all pretty straight-forward. Note that the negative acceleration (also called deceleration) has led the velocity to decrease over time. In a similar manner, a positive acceleration will cause the speed to go up. You can read more on acceleration in this blog post.

What about the distance x (in m) the object covers? We have to integrate the velocity function to find the appropriate formula. The covered distance after time t is:

2) x(t) = v(0) · t + 0.5 · a · t²

While that looks a lot more complicated, it is really just as straight-forward. Let’s go back to the car that initially has a speed of v(0) = 20 m/s and brakes with a constant a = -10 m/s². In this case the above formula becomes:

x(t) = 20 · t – 0.5 · 10 · t²

After t = 1 second, the car has traveled x(1) = 20 · 1 – 0.5 · 10 · 1² = 15 meters. By the time it comes to a halt at t = 2 seconds, it moved x(2) = 20 · 2 – 0.5 · 10 · 2² = 20 meters. Note that we don’t have to use the time as a variable. There’s a way to eliminate it. We could solve equation 1) for t and insert the resulting expression into equation 2). This leads to a formula connecting the velocity v and distance x.

3) Constant acceleration_html_b85f3ec

Solved for x it looks like this:

3)’ Constant acceleration_html_m23bb2bb3

It’s a very useful formula that you should keep in mind. Suppose a tram accelerates at a constant a = 1.3 m/s², which is also a realistic value, from rest (v(0) = 0 m/s). What distance does it need to go to full speed v = 10 m/s? Using equation 3)’ we can easily calculate this:

Constant acceleration_html_m11de6604

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Here are a few exercises and solutions using the equations 1), 2) and 3).

1. During free fall (air resistance neglected) an object accelerates with about a = 10 m/s. Suppose the object is dropped, that is, it is initially at rest (v(0) = 0 m/s).

a) What is its speed after t = 3 seconds?
b) What distance has it traveled after t = 3 seconds?
c) Suppose we drop the object from a tower that is x = 20 meters tall. At what speed will it impact the ground?
d) How long does the drop take?

Hint: in exercise d) solve equation 1) for t and insert the result from c)

2. During the reentry of space crafts accelerations can be as high as a = -70 m/s². Suppose the space craft initially moves with v(0) = 6000 m/s.

a) What’s the speed and covered distance after t = 10 seconds?
b) How long will it take the space craft to half its initial velocity?
c) What distance will it travel during this time?

3. An investigator arrives at the scene of a car crash. From the skid marks he deduces that it took the car a distance x = 55 meters to come to a halt. Assume full braking (a = -10 m/s²). Was the car initially above the speed limit of 30 m/s?

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Solutions to the exercises:

Exercise 1

a) 30 m/s
b) 45 m
c) 20 m/s
d) 2 s

Exercise 2

a) 5,300 m/s and 56,500 m
b) 42.9 s (rounded)
c) 192,860 m (rounded)

Exercise 3

Yes (he was initially going 33.2 m/s)

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To learn the basic math you need to succeed in physics, check out the e-book “Algebra – The Very Basics”. For an informal introduction to physics, check out the e-book “Physics! In Quantities and Examples”. Both are available at low prices and exclusively for Kindle.

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The 7 Laws of Customer Service

If you want to provide solid and helpful customer service, you should stick to a number of basic rules that make you stand out from the crowd. Here are 7 rules that can help in doing so:

1. Roll Out The Red Carpet For Everyone. If there’s one thing people hate about poor service, it’s getting treated differently from others. It makes them feel inferior and second-class. Gary Richter says you should roll out the red carpet for everyone, but particularly those who don’t expect it. “I tell my employees, if we roll out the red carpet for a billionaire, they won’t even notice. If we roll it out for millionaires, they expect it. If we roll it out for thousandaires, they appreciate it. And, if we roll out the red carpet for hundredaires, they’ll tell everyone they know.”

2. Take Time To Know Your Customers. The fast pace of modern living together with advances in technology have together put a non-human face on much of our customer service. If you can find a way to re-connect with your customers one-on-one, you’ll strike a chord with your customers that will be like a streak of gold. Kathy Burns remembers a time when people took time to care and listen. “Some of you may remember, and others may have heard stories about, a time in life when the doctor would come to your home to check on you if you were ill. Or maybe you’ve heard about going down to your local pharmacy and having the owner greet you by name and ask how you’re doing. Not only did they ask, but they really wanted to know the answer and they took the time to listen to what you had to say. That’s customer service – taking the time to know your customers, really caring about how they feel, and wanting to go the extra mile to make sure they’re happy.”

3. Be Easy To Do Business With. One of the problems with modern businesses is that the systems we use to save time and money are often devised for the company’s benefit and not the customers. As a result, the customer experience is frustrating and difficult. Tracey Lowrance says this needs to be reversed. “Customers expect single source service. Customers don’t want to be transferred to every unit of your business to have their problems solved. They want to be able to do business with you with the slightest amount of discomfort. You must be easy to do business with.”

4. Go Out Of Your Way To Make Sure They’re Happy. One of the most important things your customers want from you is a guarantee that your product or service will work. So move heaven and earth to make sure it does. Bob Leduc suggests you shouldn’t make people pay until they are fully happy. “Instead of offering a money back guarantee, a service business can provide a guarantee to solve the customer’s problem. For example, a plumber can guarantee to come back without charge as often as necessary to stop the leak. A landscaper can replace without charge any plants that don’t survive for at least 6 months. A sales consultant can continue working without charge until the promised sales results are achieved.”

5. Notice What Customers See. A big part of what customers think about you comes from what they see and believe. Personal Selling Power noticed the following difference in two candy stores. “Although two competing candy stores had the same prices, neighbourhood kids preferred one store to the other. When asked why, they said, “Because the person in the good store always gives us more candy. The girl in the other store takes candy away.” True? Not really. In the good store the owner would always make sure to put a small amount of candy on the scale and then keep adding to it. In the bad store, the owner would pile a heaping amount of candy on the scale, and then take it off until it hit the right weight. The same amount of candy was sold, but perception is everything.”

6. Work On Everything The Customer Experiences. The customer experience isn’t just receiving the service or buying the goods. It’s about all the other little bits and pieces in-between. Such as the manner of the receptionist, the state of the floors and tables, the attitude of other staff, the ease of parking, the tone of the notices, the smile or lack of it on the face of the checkout team. Be like the Mirage hotel in Las Vegas who have a slogan that says: “We spend 600 hours a week pampering the plants. Imagine what we’ll do for our guests.”

7. Believe In Customer Service. To become a great service organization, you have to believe in customer service from the bottom of your soul. It has to be part of the way you work. Anita Roddick, founder of retail cosmetic franchise group Body Shop puts it like this: “I am still looking for the modern equivalent of those Quakers who ran successful businesses, made money because they offered honest products and treated people decently, worked hard, spent honestly, saved honestly, gave honest value for money, put back more than they took out and told no lies. This business creed, sadly, seems long forgotten.”

If you’re interested in learning more about customer service, be sure to check out the Service Recovery Paradox.