# Mario Particle Motionap Calculus

## 2011-2012 Particle Motion Definition and Calculus

- (b) The particle moves continuously and monotonically from (0) 2x to x(3) 10. Similarly, the particle moves continuously and monotonically from x(3) 10 to x(5) 7 and also from x(5) 7 to (6) 9x. By the Intermediate Value Theorem, there are three values of t for which the particle is at ( ) 8xt.
- Calculus AB Notes on Particle Motion Particle motion describes the physics of an object (a point) that moves along a line; usually horizontal. There are 3 different functions that model this motion.

In physics and calculus courses alike, the concept of distance and displacement, and how it relates to acceleration, velocity, and position is called the study of particle motion, and utilizes the definite integral.

The position of a particle (in inches) moving along the *x*-axis after *t* seconds have elapsed is given by the following equation:

*s* = *f*(*t*) = *t*^{4} – 2*t*^{3} – 6*t*^{2} + 9*t*

(a) Calculate the velocity of the particle at time *t*.

(b) Compute the particle’s velocity at *t* = 1, 2, and 4 seconds.

(c) When is the particle at rest?

(d) When is the particle moving in the forward (positive) direction?

(e) Calculate total distance traveled by the particle (i.e., forwards and backwards) after *t* = 5 seconds.

(f) Calculate the acceleration of the particle after 4 seconds.

(g) When is the speed of the particle constant?

### Solution:

(a) The velocity is the derivative of position, so the velocity is *v*(*t*) = 4*t*^{3} – 6*t*^{2} – 12*t* + 9.

(b) Simply plug into the velocity equation to get: *v*(1) = –5 in/sec, *v*(2) = –7 in/sec, *v*(4) = 121 in/sec.

(c) If you graph the velocity function on your calculator, you see that it appears to pass through *x* = –1.5. Use synthetic division to ensure that this is true and to factor the equation. You will get the following:

(*t* – 1/2)(4*t*^{2} – 12*t* + 6)

Now, use the quadratic formula to solve the quadratic part, and you’ll see that the velocity equals zero (in other words, is stopped) when *t* = –1.5, 0.6339745962, 2.366025404. Even though you can round to the third decimal place, you need to use these values for the remainder of the problem.

(d) If you plug in values into the velocity equation between the *x*-intercepts above, you will get positive values on the intervals (–1.5, 0.6339) and (2.366, ∞). Note that it doesn’t quite make sense to have negative time, so (0, 0.6339) is just as acceptable, and perhaps more so, for the first interval. We do this because positive velocity implies forward movement.

(e) First, substitute the “turn points” you found in part (c) into the position equation. When the velocity equals zero in this problem, the particle is stopping because it is turning to go the other way. You find that*s*(0.6339) = 2.946152423,* s*(2.366) = –7.446152423, *and s*(5) = 270. Note that the negative *x*-intercept is ignored because you cannot move back in time.

These numbers represent how far the particle is from the origin at specific times. So, the particle moves 2.9 inches to the right of the origin, then moves 7.44 inches left of it, and finally ends up 270 inches to the right of it. By the time *t* = 2.366 seconds, the particle has traveled to the right 2.9 inches, back 2.9 inches to the origin, then left 7.4 more inches. It then moves 7.4 inches back to the origin and ends up 270 more inches to the right of it. The final answer is 290.785 inches.

(f) The acceleration is the derivative of velocity, so a(t)* = v’*(t) = 12*t*^{2} – 12*t* – 12. The acceleration at *t* = 4 seconds is *a*(4) = 132 in/sec^{2}.

(g) Set the acceleration equal to zero and solve using the quadratic equation:* t* = –0.618 sec or 1.618 sec.

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Have you ever wondered what the difference is between distance and **displacement**?

Look no further!

Distance is the amount traveled, regardless of any change in direction along the way, while displacement is just the amount span or interval between the starting point and ending point.

The Physics Classroom rightly compares these two ideas by stating that distance is a scalar or length, that refers to how much ground an object has covered during its motion, while displacement is a vector with length and direction, and refers to an object’s overall change in position.

Simple enough, right?

But you know what is so confusing for students? Sometimes textbooks and professors use different terminology in expressing length and motion!

In some instances, the word distance will be used to describe displacement, and the phrase total distance will be used in place of the word distance.

So how will we know the difference?

There are some easy tricks for figuring out what a question is talking about, and this video will clear up any confusion.

In physics and calculus courses alike, the concept of distance and displacement, and how it relates to acceleration, velocity, and position is called the study of particle motion, and utilizes the definite integral.

We will draw upon our previous knowledge of how to find critical numbers to determine when a particle is at rest and if/when it changes direction.

Then we will learn how to write definite integrals given an initial condition and a derivative expression, and use the integral expression to find both distance and displacement.

Finally, we will look at a Free Response Question where we will use most of our derivative and integral techniques, as this question will ask us to find the:

- instantaneous rate of change
- average rate of change
- increasing and decreasing intervals
- definite integral expression for displacement
- average value function
- total distance

## Mario Particle Motion Ap Calculus Problems

## Particle Motion Video

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## Mario Particle Motion Ap Calculus Notes

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