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## Physics library

### Course: Physics library>Unit 1

Lesson 3: Acceleration

# What is acceleration?

Velocity describes how position changes; acceleration describes how velocity changes. Two layers of change!

## What does acceleration mean?

Compared to displacement and velocity, acceleration is like the angry, fire-breathing dragon of motion variables. It can be violent; some people are scared of it; and if it's big, it forces you to take notice. That feeling you get when you're sitting in a plane during take-off, or slamming on the brakes in a car, or turning a corner at a high speed in a go kart are all situations where you are accelerating.
Acceleration is the name we give to any process where the velocity changes. Since velocity is a speed and a direction, there are only two ways for you to accelerate: change your speed or change your direction—or change both.
If you’re not changing your speed and you’re not changing your direction, then you simply cannot be accelerating—no matter how fast you’re going. So, a jet moving with a constant velocity at 800 miles per hour along a straight line has zero acceleration, even though the jet is moving really fast, since the velocity isn’t changing. When the jet lands and quickly comes to a stop, it will have acceleration since it’s slowing down.
Or, you can think about it this way. In a car you could accelerate by hitting the gas or the brakes, either of which would cause a change in speed. But you could also use the steering wheel to turn, which would change your direction of motion. Any of these would be considered an acceleration since they change velocity.

## What's the formula for acceleration?

To be specific, acceleration is defined to be the rate of change of the velocity.
a, equals, start fraction, delta, v, divided by, delta, t, end fraction, equals, start fraction, v, start subscript, f, end subscript, minus, v, start subscript, i, end subscript, divided by, delta, t, end fraction
The above equation says that the acceleration, a, is equal to the difference between the initial and final velocities, v, start subscript, f, end subscript, minus, v, start subscript, i, end subscript, divided by the time, delta, t, it took for the velocity to change from v, start subscript, i, end subscript to v, start subscript, f, end subscript.
Note that the units for acceleration are start fraction, start text, m, end text, slash, s, divided by, start text, s, end text, end fraction , which can also be written as start fraction, start text, m, end text, divided by, start text, s, end text, squared, end fraction. That's because acceleration is telling you the number of meters per second by which the velocity is changing, during every second. Keep in mind that if you solve a, equals, start fraction, v, start subscript, f, end subscript, minus, v, start subscript, i, end subscript, divided by, delta, t, end fraction for v, start subscript, f, end subscript, you get a rearranged version of this formula that’s really useful.
v, start subscript, f, end subscript, equals, v, start subscript, i, end subscript, plus, a, delta, t
This rearranged version of the formula lets you find the final velocity, v, start subscript, f, end subscript, after a time, delta, t, of constant acceleration, a.

I have to warn you that acceleration is one of the first really tricky ideas in physics. The problem isn’t that people lack an intuition about acceleration. Many people do have an intuition about acceleration, which unfortunately happens to be wrong much of the time. As Mark Twain said, “It ain’t what you don’t know that gets you into trouble. It’s what you know for sure that just ain’t so.”
The incorrect intuition usually goes a little something like this: “Acceleration and velocity are basically the same thing, right?” Wrong. People often erroneously think that if the velocity of an object is large, then the acceleration must also be large. Or they think that if the velocity of an object is small, it means that acceleration must be small. But that “just ain’t so”. The value of the velocity at a given moment does not determine the acceleration. In other words, I can be changing my velocity at a high rate regardless of whether I'm currently moving slow or fast.
To help convince yourself that the magnitude of the velocity does not determine the acceleration, try figuring out the one category in the following chart that would describe each scenario.
high speed, low acceleration A car flooring it out of a red light A car that is driving at a slow and nearly steady velocity through a school zone A car that is moving fast and tries to pass another car on the freeway by flooring it A car driving with a high and nearly steady velocity on the freeway

I wish I could say that there was only one misconception when it comes to acceleration, but there is another even more pernicious misconception lurking here—it has to do with whether the acceleration is negative or positive.
People think, “If the acceleration is negative, then the object is slowing down, and if the acceleration is positive, then the object is speeding up, right?” Wrong. An object with negative acceleration could be speeding up, and an object with positive acceleration could be slowing down. How is this so? Consider the fact that acceleration is a vector that points in the same direction as the change in velocity. That means that the direction of the acceleration determines whether you will be adding to or subtracting from the velocity. Mathematically, a negative acceleration means you will subtract from the current value of the velocity, and a positive acceleration means you will add to the current value of the velocity. Subtracting from the value of the velocity could increase the speed of an object if the velocity was already negative to begin with since it would cause the magnitude to increase.
If acceleration points in the same direction as the velocity, the object will be speeding up. And if the acceleration points in the opposite direction of the velocity, the object will be slowing down. Check out the accelerations in the diagram below, where a car accidentally drives into the mud—which slows it down—or chases down a donut—which speeds it up. Assuming rightward is positive, the velocity is positive whenever the car is moving to the right, and the velocity is negative whenever the car is moving to the left. The acceleration points in the same direction as the velocity if the car is speeding up, and in the opposite direction if the car is slowing down.
Another way to say this is that if the acceleration has the same sign as the velocity, the object will be speeding up. And if the acceleration has the opposite sign as the velocity, the object will be slowing down.

## What do solved examples involving acceleration look like?

### Example 1:

A neurotic tiger shark starts from rest and speeds up uniformly to 12 meters per second in a time of 3 seconds.
What was the magnitude of the average acceleration of the tiger shark?
a, equals, start fraction, v, start subscript, f, end subscript, minus, v, start subscript, i, end subscript, divided by, delta, t, end fraction
Plug in the final velocity, initial velocity, and time interval.
a, equals, start fraction, 12, start fraction, start text, m, end text, divided by, start text, s, end text, end fraction, minus, 0, start fraction, start text, m, end text, divided by, start text, s, end text, end fraction, divided by, 3, start text, s, end text, end fraction
Calculate and celebrate!
a, equals, 4, start fraction, start text, m, end text, divided by, start text, s, end text, squared, end fraction

### Example 2:

A bald eagle is flying to the left with a speed of 34 meters per second when a gust of wind blows back against the eagle causing it to slow down with a constant acceleration of a magnitude 8 meters per second squared.
What will the speed of the bald eagle be after the wind has blown for 3 seconds?
a, equals, start fraction, v, start subscript, f, end subscript, minus, v, start subscript, i, end subscript, divided by, delta, t, end fraction
Symbolically solve to isolate the final velocity on one side of the equation.
v, start subscript, f, end subscript, equals, v, start subscript, i, end subscript, plus, a, delta, t
Plug in the initial velocity as negative since it points left.
v, start subscript, f, end subscript, equals, minus, 34, start fraction, start text, m, end text, divided by, start text, s, end text, end fraction, plus, a, delta, t
Plug in acceleration with opposite sign as velocity since the eagle is slowing.
v, start subscript, f, end subscript, equals, minus, 34, start fraction, start text, m, end text, divided by, start text, s, end text, end fraction, plus, 8, start fraction, start text, m, end text, divided by, start text, s, end text, squared, end fraction, delta, t
Plug in the time interval during which the acceleration acted.
v, start subscript, f, end subscript, equals, minus, 34, start fraction, start text, m, end text, divided by, start text, s, end text, end fraction, plus, 8, start fraction, start text, m, end text, divided by, start text, s, end text, squared, end fraction, left parenthesis, 3, start text, s, end text, right parenthesis
Solve for the final velocity.
v, start subscript, f, end subscript, equals, minus, 10, start fraction, start text, m, end text, divided by, start text, s, end text, end fraction
The question asked for speed; since speed is always a positive number, the answer must be positive.
start text, f, i, n, a, l, space, s, p, e, e, d, end text, equals, plus, 10, start fraction, start text, m, end text, divided by, start text, s, end text, end fraction
Note: Alternatively we could have taken the initial direction of the eagle's motion to the left as positive, in which case the initial velocity would have been plus, 34, start fraction, start text, m, end text, divided by, start text, s, end text, end fraction, the acceleration would have been minus, 8, start fraction, start text, m, end text, divided by, start text, s, end text, squared, end fraction, and the final velocity would have come out to equal plus, 10, start fraction, start text, m, end text, divided by, start text, s, end text, end fraction. If you always choose the current direction of motion as positive, then an object that is slowing down will always have a negative acceleration. However, if you always choose rightward as positive, then an object that is slowing down could have a positive acceleration—specifically, if it is moving to the left and slowing down.

## Want to join the conversation?

• Can't something change direction and not be accelerating?? • Can someone please give the correct answers for the car exercise? I don't understand the explanation. and what is exactly meant by flooring in? • Could someone re-explain the picture with the four cars? I'm not quite sure about why the car slows down if the signs of velocity and acceleration are oppposite and why it speeds up when they have the same signs.
Actually, the first two pictures where the velocity is positive is quite simple to get, but the cases that confuse me the most are the ones in which the velocity is negative. • Good, clear question.

Maybe this helps...

F=ma right? so if they are equal, they are in the same direction. Kinda makes sense I guess: if you push something, it will accelerate in that same direction.
So, if you are ok with that idea, think about the acceleration arrow (vector) in your diagrams as being replaced or equal to the force vector.

Does that help or no??
• So when we accelerate, we feel as if a force was dragging us backwards and let us feel the inertia. Can we use this principle to make some thing which measures acceleration? • Probably no.

And the force that drag us backward IS inertia. (Sorry if I'm mistaken. You seemed to treat inertia and 'the force that drag us backwards' as a separate thing.)

Inertia is the tendency of an object to remain in its state. For example, a rolling ball will continue rolling in a straight line due to its inertia unless compelled by an external force. (friction, gravity, somebody kicking it etc) Same thing, inertia keeps a ball stationary unless someone apply force on it.
And to correspond to your given example, I think the bus and passenger situation will be a better explanation:
Passengers tend to move backwards when a bus accelerates. This is because the inertia in their body keep them at rest. So, passengers will lean backwards when the bus moves forward.

Well, inertia is a measure of mass though.

Hopes this helps.
• What is the main or basic difference between speed and velocity? Acceleration is used only for change in velocity or even for a change in speed? • Speed is the magnitude of velocity. Velocity is a vector, which means it has two parts: first, your velocity has a magnitude, which just answers the question "how fast?", but does not say anything about the direction the object is moving. The second part of velocity is its direction, which answers the question "which way?". So a velocity might be "20 m/s, downward". The speed is 20 m/s, and the direction is "downward". Acceleration is the rate of change of velocity. Usually, acceleration means the speed is changing, but not always. When an object moves in a circular path at a constant speed, it is still accelerating, because the direction of its velocity is changing.
• out of curiosity. While slowing down, why should it be called as negative acceleration rather than deceleration? • I don't understand: How does -34m/s+8m/s^2(3s)= -10m/s? • Your current velocity is 34 m/s in the opposite direction of what is considered the "positive" direction, so it is -34 m/s. Acceleration, 8 m/s^2, is the change in velocity, and in this case it is in the positive direction. So, the velocity will become 8 m/s more positive for every second that this acceleration is present. (8 m/s^2)*(3s)=24 m/s, This is a positive change in velocity, so -34 m/s + 24 m/s=-10 m/s. You are still moving in the opposite direction but at a slower magnitude.
• how do u determine if the acceleration is positive or negative? • can someone explain how the units for the final solved example went back to m/s please? • In the final solved example, the final answer found is velocity, not acc. Units of velocity are m/s.

The equation was Vf = Vi + at. Now we know units of Vf and Vi will be m/s, since they're both velocities.
Also, we know that acc. 'a'=V/t(Change in velocity over time). We can write this as V=at(multiplying both sides by t). So 'at' is essentially a velocity and hence will have the same units as velocity, which is m/s.

Also, you can only add quantities with the same units, and the sum retains the same units. E.g.: 5Litres + 3Litres gives 8Litres. It wouldn't make sense to add 5Litres to 7kilograms. If there was something I missed to explain, feel free to ask a follow-up.
Cheers! 