8.01SC | Fall 2016 | Undergraduate

# Classical Mechanics

## Week 1: Kinematics

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### Week 1 Problem Set

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Distance: Is the length of the path travelled by an object between two points in space. From its definition, the distance is a scalar and it is always a positive quantity.

Displacement: Is the change in the position of an object. If at time $$t=t_1$$ the object is at position $$\vec{r}(t_1)$$, and at a later time $$t=t_2 > t_1$$ the object is at position $$\vec{r}(t_2)$$, the displacement vector is defined as $$\Delta \vec{r} = \vec{r}(t_2) - \vec{r}(t_1)$$. In one dimension, the displacement vector has one component. For example, if the motion is along the x-axis, the displacement vector becomes $$\Delta \vec{r} = \Delta x \hat{i} = (x(t_2) - x(t_1))\hat{i}$$. The component of the displacement vector can be positive, when the final position is larger than the initial one. It can be negative, when the final position is smaller than the initial one. It can aslo be zero, if the object ends at the starting point.

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List of useful derivatives:

• Derivative of a polynomial function:

If $$\displaystyle x(t)=At^n \Longrightarrow \frac{dx}{dt}=nAt^{n-1}$$

where $$A$$ and $$n$$ are constants.

• Derivative of an exponential function:

If $$\displaystyle x(t)=A e^{bt} \Longrightarrow \frac{dx}{dt}=Ab e^{bt}$$

where $$A$$ and $$b$$ are constants.

• Derivative of a logarithmic function:

If $$\displaystyle x(t)=A\ln(b+ct) \Longrightarrow \frac{dx}{dt}=\frac{Ac}{b+ct}$$

where $$A$$, $$b$$ and $$c$$ are constants.

• Derivative of sine:

If $$\displaystyle x(t)=A\sin(b+ct) \Longrightarrow \frac{dx}{dt}=Ac \cos(b+ct)$$

where $$A$$, $$b$$ and $$c$$ are constants.

• Derivative of cosine:

If $$\displaystyle x(t)=A\cos(b+ct) \Longrightarrow \frac{dx}{dt}=-Ac \sin(b+ct)$$

where $$A$$, $$b$$ and $$c$$ are constants.

External References

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A home-built model rocket is launched straight up into the air. At time $$\displaystyle t = 0$$ the rocket is at rest, about to be launched. The position of the rocket as a function of time is given by:

$$\displaystyle y(t) = \frac{1}{2}(a_0-g)t^2 - \frac{1}{30}\frac{a_0 }{t_{0}^{4}}t^{6}$$ for $$\displaystyle 0 \lt t \lt t_0$$

where $$\displaystyle a_0$$ is a positive constant, $$\displaystyle g$$ is the acceleration of gravity and $$\displaystyle a_0 > g$$. The contant $$\displaystyle t_0$$ is the amount of time that the fuel takes to burn out. Express your answer in terms of g, t,$$\displaystyle a_0$$, and $$\displaystyle t_0$$.

Find $$\displaystyle a$$, the $$\displaystyle y$$-component of the acceleration as a function of time.

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A runner travels along the x-axis, and at time $$\displaystyle t=0$$ is at the origin. The $$\displaystyle x$$-component of the runner’s position with respect to the origin is given by:

$$\displaystyle x(t)=bt^2$$

where $$\displaystyle b$$ is a positive constant.

(Part a) What are the units of the constant $$\displaystyle b$$? Express your answer in terms of m for meter and s for seconds.

(Part b) Find $$\displaystyle v(t)$$, the $$\displaystyle x$$-component of the runner’s velocity as a function of time.

(Part c) Find $$\displaystyle a(t)$$, the $$\displaystyle x$$-component of the runner’s acceleration as a function of time.

(Part d) Do a plot of $$\displaystyle x(t)$$, $$\displaystyle v(t)$$ and $$\displaystyle a(t)$$ vs. time.

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List of useful integrals

• Integral of a polynomial function:

If $$x(t)=At^n \Longrightarrow \int_{t_{i}}^{t_{f}} x(t)dt=\frac{A}{n+1}(t_{f}^{n+1}-t_{i}^{n+1})$$

where $$A$$ and $$n$$ are constants.

• Integral of an exponential function:

If $$x(t)=A e^{bt}\Longrightarrow \int_{t_{i}}^{t_{f}} x(t)dt=\frac{A}{b}(e^{bt_{f}}-e^{bt_{i}})$$

where $$A$$ and $$b$$ are constants.

• Integral of $$1/t$$:

If $$x(t)=\frac{1}{t} \Longrightarrow \int_{t_{i}}^{t_{f}} x(t)dt=\ln(t_{f})-\ln(t_{i})=\ln(\frac{t_{f}}{t_{i}})$$

• Integral of sine:

If $$x(t)=A\sin(b+ct) \Longrightarrow \int x(t)dt=-\frac{A}{c}\cos(b+ct) + D$$

where $$A$$, $$b$$ and $$c$$ are constants, and $$D$$ is an integration constant.

• Integral of cosine:

If $$x(t)=A\cos(b+ct) \Longrightarrow \int x(t)dt=\frac{A}{c} \sin(b+ct) + D$$

where $$A$$, $$b$$ and $$c$$ are constants, and $$D$$ is an integration constant.

External References

Wolfram Alpha

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A special case of Two Dimensional Motion is the motion of an object with an initial velocity with a non-zero horizontal component under the influence of only the gravitational force. This motion is referred to as Projectile Motion.

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### Demo: Relative Motion Gun

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Problem Set 1 contains the following problems:

1. Car and Bicycle Rider
2. Elevator Trip
3. Rocket Launch
4. Throw and Catch
5. Vertical Collision

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This sequence will provide some guided problems and example problems. We start with a brief note about problem solving strategies.

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An apple is suspended a height $$\displaystyle h$$ above the ground. A physics demo instructor has set up a projectile gun a horizontal distance $$\displaystyle d$$ away from the apple. The projectile is initially a height $$\displaystyle s$$ above the ground. The demo instructor fires the projectile with an initial velocity of magnitude $$\displaystyle v_0$$ just as the apple is released. Find $$\displaystyle \theta _0$$, the angle at which the projectile gun must be aimed in order for the projectile to strike the apple. Ignore air resistance.

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At time $$\displaystyle t=0$$, a car moving along the +$$\displaystyle x$$-axis passes through $$\displaystyle x=0$$ with a constant velocity of magnitude $$\displaystyle v_0$$. At some time later, $$\displaystyle t_1$$, it starts to slow down. The acceleration of the car as a function of time is given by:

 $$\displaystyle a(t) = \left\{ \begin{array}{ll} 0 & \quad 0 \leq t \leq t_1 \\ -c(t-t_1) & \quad t_1 \lt t \leq t_2 \end{array} \right.$$

where $$\displaystyle c$$ is a positive constants in SI units, and $$\displaystyle t_1 \lt t \leq t_2$$ is the given time interval for which the car is slowing down. The goal of the problem is to find the car’s position as a function of time between $$\displaystyle t_{1} \lt t \lt t_2$$. Express your answer in terms of v_0 for $$\displaystyle v_0$$, t_1 for $$\displaystyle t_1$$, t_2 for $$\displaystyle t_2$$, and $$\displaystyle c$$ as needed.

(Part a). What is $$\displaystyle v(t)$$, the velocity of the car as a function of time during the time interval $$\displaystyle 0 \leq t \leq t_1$$?

(Part b). What is $$\displaystyle x(t)$$, the position of the car as a function of time during the time interval $$\displaystyle 0 \leq t \leq t_1$$?

(Part c). What is $$\displaystyle v(t)$$, the velocity of the car as a function of time during the time interval $$\displaystyle t_1 \lt t \leq t_2$$?

(Part d). What is $$\displaystyle x(t)$$, the position of the car as a function of time during the time interval $$\displaystyle t_1 \lt t \leq t_2$$?

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A particle of mass $$\displaystyle m$$ is moving along the +$$\displaystyle x$$-axis with a constant speed $$\displaystyle v_0$$. At time $$\displaystyle t = 0$$ it enters a region in space between two parallel plates which are separated by a distance $$\displaystyle h$$ and are contained in the $$\displaystyle x$$-$$\displaystyle z$$ plane. The figure shows a side view of the plates, the +$$\displaystyle z$$ axis is out of the screen. While the particle is between the plates, it is acted on by both gravity and by a time varying force that points upward (along the +$$\displaystyle y$$-direction) and has magnitude $$\displaystyle F = bt$$, where $$\displaystyle b$$ is a positive constant that is sufficiently large such that the particle hits the top plate without ever touching the bottom plate.

(Part 1) What are the units of the constant $$\displaystyle b$$? Enter m for meters, s for seconds and kg for kilograms. Make a sketch of what you think the trajectory of the particle is as it moves through the plates. Draw a coordinate system showing the position of the particle at time $$\displaystyle t$$. Clearly indicate your origin, choice of axis, and draw in the coordinate functions for the position of the particle at time $$\displaystyle t$$?

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You are in a car standing by a traffic light and at time $$\displaystyle t=0$$ the light turns green. You start to accelerate during the first $$\displaystyle t_1$$ seconds so that the acceleration of your car is given by:

 $$\displaystyle a_1(t) = \left\{ \begin{array}{ll} b_1 & \quad 0 \leq t \leq t_1 \\ 0 & \quad t_1 < t \leq t_2 \end{array} \right.$$

where $$\displaystyle b_1$$ is a positive constant.

At the instant the light turns green a cyclist passes through the intersection moving with a speed $$\displaystyle v_0$$ in the same direction as your car is moving. At that instant, the cyclist starts to brake with a constant acceleration of magnitude $$\displaystyle b_2$$. At time $$\displaystyle t = t_2$$ the cyclist stops at the same location where you are.

The goal of the problem is to calculate the value of $$\displaystyle b_2$$ in terms of the given variables, $$\displaystyle b_1$$, $$\displaystyle v_0$$, $$\displaystyle t_1$$ and $$\displaystyle t_2$$.

First: Describe the motion of the car. Given the car’s acceleration $$\displaystyle a_1(t)$$, find its velocity and its position as a function of time:

(Part b) Calculate $$\displaystyle x_1(t)$$ , the car’s position as a funcion of time. Express your answer in terms of $$\displaystyle t$$, $$\displaystyle t_1$$, and $$\displaystyle b_1$$ as needed.

Second: Describe the motion of the bicycle. Given the bicycle’s acceleration $$\displaystyle a_2(t)$$, find its velocity and its position:

(Part c) Calculate $$\displaystyle v_2(t)$$ , the bicycle’s velocity as a funcion of time. Express your answer in terms of $$\displaystyle t$$, $$\displaystyle b_2$$, $$\displaystyle v_0$$ as needed.

(Part d) Calculate $$\displaystyle x_2(t)$$ , the bicycle’s position as a funcion of time. Express your answer in terms of $$\displaystyle t$$,$$\displaystyle b_2$$, and $$\displaystyle v_0$$ as needed.

Third: Find the value of $$\displaystyle b_2$$. We know that the bicycle stops at $$\displaystyle t=t_2$$, this condition is expressed as:

$$\displaystyle v_2(t_2) = 0$$ (eq. 1)

We also know that the bicycle and the car are at the same location when the bicycle stops. This condition implies:

$$\displaystyle x_2(t_2) = x_1(t_2)$$ (eq. 2)

Use (eq.1) and (eq. 2) to obtain the value of $$\displaystyle b_2$$. Express your answer in terms of $$\displaystyle t_1$$, $$\displaystyle b_1$$, and $$\displaystyle v_0$$. Do not use $$\displaystyle t_2$$ in your answer.

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Fall 2016
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