Linear Acceleration

by Jason "Mussolini" Crandall and Joe "Stalin" Thaler, December 1997

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1.

A car in front of the school goes from rest to 27 m/s in 3.0 seconds. What is its acceleration? Here is what you start with:

```
t  = 3.0 s
```

V_o = 0 m/s (assumed because it starts from rest)

```
V  = 27 m/s
```
```
a  = ?
```

Once again, our formula is:

V = V_o + at

Plug in the initial and final velocities and the time and calculate the acceleration. 

27 m/s = 0 m/s + (a * 3.0 seconds) 

Our solution is 9 m/s/s. 

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2.

A train can speed up at a rate of .053 m/s/s. What time will it take for it to reach a speed of 21 m/s from a standing stop?
Here is what you start with:

```
t  = ?
```
```
V_o =  0 (assumed)
```
```
V  = 21 m/s
```
```
a  = .053 m/s/s
```

Once again, our formula is:

V = V_o + at

Thus:

21 m/s = 0 + (.053 m/s/s * t)

Solve for t and you get 396 seconds.

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3.

A rocket accelerates at a rate of 190 m/s/s for 2.4 seconds from rest.
What is its final speed? (456 m/s)
Here is what you start with:

```
t  =2.4 s
```
```
V_o =  0 (assumed)
```
```
V  = ?
```
```
a  = 190 m/s/s
```

You can use  V = V_o + at  to find the final velocity:

V = 0 + (190 m/s/s)(2.4 s) = 456 m/s

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4.

A car has a velocity of +15 m/s. It then accelerates at a rate of +3.5 m/s/s for the next 5 seconds. What is its final velocity? What is your favorite ice cream flavor (chocolate and vanilla do not apply)? (32.5 m/s, pralines and cream)
Here is what you start with:

```
t  = 5 s
```
```
V_o = 15 m/s
```
```
V  = ?
```
```
a  = +3.5 m/s/s
```

Ice Cream = Anything but chocolate and vanilla

Use  V = V_o + at  to find the final Velocity:

V = 15 m/s + (3.5 m/s/s)(5 s) = 32.5 m/s

Now the ice cream part is a little bit more tricky.  Without any formulas, we are forced to

rely upon empirical observation.  Therefore I took it upon myself to sample all 31 flavors

at Baskin-Robbins.  One empty wallet and several ice cream headaches later, I have come to

the conclusion that Pralines and Cream is the favorite ice cream flavor.  Judging from my

experience, attempts to reproduce this experiment may be hazardous to your health and

financial well-being.

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5.

A car involved in an 14 m/s collision with a parked car is determined to have skidded for a time of 3.7 seconds before the impact. If that particular car can brake at -8.2 m/s/s with the tires locked, how fast was the car going before it hit its brakes? (44 m/s)
Here is what you start with:

```
t  = 3.7 s
```
```
V_o = ?  
```
```
V  = 14 m/s (at impact)
```
```
a  = -8.2 m/s/s
```

Use  V = V_o + at  to find the initial Velocity:

14 m/s = V_o + (-8.2 m/s/s * 3.7 s)

Thus (use a little algebra), V_o is 44 m/s. 44 m/s is approximately 98 mph, so our

little friend probably deserves whatever he gets. 

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6.

A spaceship with a velocity of +320 m/s fires its retro rockets to slow it at a rate of -22 m/s/s for 3 seconds. How fast is the rocket ship going after that? (254 m/s)
Here is what you start with:

```
t  = 3 s
```
```
V_o = 320 m/s
```
```
V  = ?
```
```
a  = -22 m/s/s
```

Using  V = V_o + at  to find the final velocity:

V = 320 + (-22 m/s/s * 3 s) = 254 m/s

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7.

What time will it take a car that can accelerate from 0 m/s to 20 m/s in 5 seconds to speed up from 12 m/s to 27 m/s? (Find the acceleration) (3.75 s)
Here we have two problems:

```
t  = ?
```
```
V_o = 12 m/s
```
```
V  = 27 m/s
```
```
a  = ?
```

This is not yet solvable, because we don't know the acceleration.  

Let's assume that the acceleration from 12 m/s to 27 m/s is the same

as it is from 0 to 20 m/s.  In order to find 

the acceleration we use our

favorite formula again:  V = V_o + at.

20 m/s = 0 m/s + (a * 5 s) (Look at the first 3 numbers 

and we get a = 4 m/s/s

We now know:

```
t  = ?
```
```
V_o = 12 m/s
```
```
V  = 27 m/s
```
```
a  = 4 m/s/s
```

Now find the time using the our favorite formula once again: V = V_o + at.

27 m/s = 12 m/s + (4 m/s/s * t)

Thus, solving for t, we arrive at the answer:  3.75 seconds.

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8.

A baseball can change its velocity from +45 m/s to -52 m/s in the .015 seconds that it takes to hit a line drive. What is the acceleration of the ball? (-6500 m/s/s Rounded)
Here is what you start with:

```
t  = .015 s
```
```
V_o = +45 m/s  
```
```
V  = -52 m/s
```
```
a  = ?
```

Find the acceleration using the our freindly neighborhood aceleration formula once again:

V = V_o + at.

Plug in a couple of numbers and get:

-52 m/s = 45 m/s + (a * .015 s)

Solve for a, and we get approximately -6466.66 m/s/s, which, in case you didn't know, is 

really, REALLY fast. Using sig figs, we get -6500 m/s/s.

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9.

What must be your acceleration if you change your velocity from 34 m/s to 21 m/s in 15 seconds? (-.87 m/s/s)
Here is what you start with:

```
t  = 15 s
```
```
V_o = 34 m/s
```
```
V  = 21 m/s
```
```
a  = ?
```

Okay boys and girs, let's see if we can possibly guess what formula we're going to use today.

What's that I hear you saying?  Very good!  It IS: V = V_o + at.

Let's go to the numbers:

21 m/s = 34 m/s + (a * 15 s)

After a random survey, the majority of people polled feel that -.87 m/s/s is the correct answer.

However, 9% of our respondents DO NOT like Pralines and Cream.

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10.

Objects accelerate downwards at 32 f/s/s near the surface of the earth. For how much time must you fall to reach 60 miles/hour? Convert! (2.75 s)
Here is what you start with:

```
t  = 
```
```
V_o = 0 (assumed)
```
```
V  = 60 miles per hour = 88 f/s
```
```
a  = 32 f/s/s
```

Okay, I'm getting kind of tired of restating our favorite formula, so here it is with the

numbers already plugged in:

88 f/s = 0 + (32 f/s/s * t)

Me:   I'll take acceleration for $500 Alex.

Alex: And the answer is, 88 f/s divided by 32 f/s/s.

Me:   What is 2.75 seconds?

Alex: That's right! You win, now get off the web and go do something useful!

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