Solution:

Part A

The initial velocity of the pot is zero. Find the velocity $\displaystyle v_b$ of the pot at the bottom of the window. Then using the kinematic equation that relates initial and final velocities, acceleration, and distance traveled, you can solve for the distance $\displaystyle h$.

The average velocity of the flower pot as it passes by the window is $\displaystyle v_{avg}=\frac{L_w}{t}$.

As the pot falls past your window, there will be some instant when the pot’s velocity equals the average velocity $\displaystyle v_{avg}$. Recall that, under constant acceleration, velocity changes linearly with time. This means that the average velocity during a time interval will occur at the middle of that time interval. Meaning, the average velocity happened at time, $\displaystyle \frac{t}{2}$.

Considering the motion at the middle and at the bottom of the window.

$\displaystyle g=\frac{v_b-v_{avg}}{\frac{t}{2}}$

$\displaystyle v_b=\frac{gt}{2}+v_{ave}$

$\displaystyle v_b=\frac{gt}{2}+\frac{L_w}{t}$

So, we now know that velocity at the bottom of the window. Consider the motion from the top (point of dropped) to the bottom of the window

$\displaystyle \left(v_b\right)^2-\left(v_0\right)^2=2g\left(h-0\right)$

$\displaystyle \left(v_b\right)^2=2gh$

$\displaystyle h=\frac{\left(v_b\right)^2}{2g}$

$\displaystyle h=\frac{\left(\frac{gt}{2}+\frac{L_w}{t}\right)^2}{2g}=\frac{\left(\frac{gt^2+2L_w}{2t}\right)^2}{2g}=\frac{\left(gt^2+2L_w\right)^2}{8gt^2}$

Part B

$\displaystyle \left(v_{ground}\right)^2-\left(v_b\right)^2=2gh_b$

$\displaystyle \left(v_{ground}\right)^2=2gh_b+\left(v_b\right)^2$

$\displaystyle\left(v_{ground}\right)=\sqrt{2gh_b+\left(v_b\right)^2}$

To demonstrate the tremendous acceleration of a top fuel Drag Racer| University Physics

PART A. What is $\displaystyle t_{max}$, the longest time after the dragster begins to accelerate that you can possibly run into the back of the dragster if you continue at your initial velocity?

$\displaystyle t_{max}=\frac{v_0}{a}$

PART B. Assuming that the dragster has started at the last instant possible (so your front bumper almost hits the rear of the dragster at $\displaystyle t=t_{max}$), find your distance from the dragster when he started. If you calculate positions on the way to this solution, choose coordinates so that the position of the drag car is 0 at t=0. Remember that you are solving for a distance (which is a magnitude, and can never be negative), not a position (which can be negative).

$\displaystyle D_{start}=\frac{\left(v_0\right)^2}{2a}$

$\displaystyle D_{start}=D_{car}-x_d\left(t_{max}\right)$

$\displaystyle D_{start}=v_0t_{max}-\frac{1}{2}a\left(t_{max}\right)^2$

$\displaystyle D_{start}=v_0\left(\frac{v_0}{a}\right)-\frac{1}{2}a\left(\frac{v_0}{a}\right)^2$

$\displaystyle D_{start}=\frac{\left(v_0\right)^2}{a}-\frac{\left(v_0\right)^2}{2a}$

$\displaystyle D_{start}=\frac{\left(v_0\right)^2}{2a}$

PART C. Find numerical values for $\displaystyle t_{max}$ and $\displaystyle D_{start}$ in seconds and meters for the (reasonable) values $\displaystyle v_0=60\:mph$ (26.8 m/s) and $\displaystyle a=50\:m/s^2$

$\displaystyle t_{max}=\frac{v_0}{a}=\frac{26.8\:m/s}{50\:m/s^2}=0.54\:s$

$\displaystyle D_{start}=\frac{\left(v_0\right)^2}{2a}=\frac{\left(26.8\:m/s\right)^2}{2\left(50\:m/s^2\right)}=7.2\:m$

The blue curve shows how the car, initially at $\displaystyle x_0$, continues at constant velocity (blue) and just barely touches the accelerating drag car (red) at $\displaystyle t_{max}$.

Running and Walking Problem| University Physics

PART A. How long does it take Rick to cover the distance D?

Find the time that it takes Rick to walk the first half of the distance, that is, to travel a distance D/2 at speed $\displaystyle v_w$.

$\displaystyle t_{w,R}=\frac{D}{2v_w}$

Now find the time Rick spends running.

$\displaystyle t_{r,R}=\frac{D}{2v_r}$

Now just add the two times up and you’re done.

$\displaystyle t_R=\frac{D}{2v_w}+\frac{D}{2v_r}=\frac{D}{2v_wv_r}\left(v_w+v_r\right)$

PART B. Find Rick’s average speed for covering the distance D.

You were given the total distance and have calculated the total time. Recall that average speed is equal to the total distance traveled divided by the amount of time it took to travel this distance.

$\displaystyle v_{ave,\:R}=\frac{2v_rv_w}{v_w+v_r}$

PART C. How long does it take Tim to cover the distance?

Tim walks at speed $\displaystyle v_w$ half the time and runs at speed $\displaystyle v_r$ for the other half.

$\displaystyle v_{ave,\:T}=\frac{v_w+v_r}{2}$

The time is just the distance divided by the average speed.

$\displaystyle t_T=\frac{D}{\frac{v_w+v_r}{2}}=\frac{2D}{v_r+v_w}$

PART D. Who covers the distance D more quickly?

Imagine that both Rick and Tim do all of their walking before they start to run. Rick will start running when he has covered half of the total distance. When Tim reaches half of the total distance, will he already have started running?

PART E. In terms of given quantities, by what amount of time, Δt, does Tim beat Rick?

$\displaystyle \Delta t=\frac{D\left(v_w-v_r\right)^2}{2v_rv_w\left(v_r-v_w\right)}$

This is just simple subtraction between the two computed times.

PART F. In the special case that vr=vw, what would be Tim’s margin of victory Δt(vr=vw)?

If vr=vw, is the any difference between what Tim and Rick do?

Half the Distance and Half the Time Problem| University Physics

PART A. What is Julie’s average speed on the way to Grandmother’s house?

Julie drove 50 miles at a speed of 35 mph, and drove another 50 miles for 65 mph. So, for the first 50 miles, she drove for

$\displaystyle time=\frac{distance}{speed}=\frac{50\:mi}{35\:mph}=\frac{10}{7}\:hours$

and for the next 50 miles, she drove for

$\displaystyle time=\frac{distance}{speed}=\frac{50\:mi}{65\:mph}=\frac{10}{13}\:hours$

Therefore, her average speed was

$\displaystyle average\:speed=\frac{total\:distance}{total\:time}$

$\displaystyle average\:speed=\frac{100\:miles}{\frac{10}{7}+\frac{10}{13}\:hours}=45.5\:mph$

PART B. What is her average speed on the return trip?

Since the time she used driving at 35 mph is the same amount of time she used driving at 65 mph, the average speed is just the average of the two speeds given.

Position, Velocity, and Acceleration values from Position Function| University Physics

PART A.

Evaluate the position at time t= 3.00 s.

$\displaystyle x=2.00\left(3\right)^3-5.00\left(3\right)+3=42.0\:m$

PART B.

Determine the velocity function v(t) from the position function x(t) by differentiation.

$\displaystyle v\left(t\right)=6t^2-5$

$\displaystyle v\left(t\right)=6\left(3\right)^2-5=49.0\:m/s$

PART C.

Determine the acceleration function a(t) from the the velocity function by differentiation

$\displaystyle a\left(t\right)=12t$

$\displaystyle a\left(t\right)=12t=12\left(3\right)=36\:m/s^2$

Analyzing Position versus Time Graphs: Conceptual Question| University Physics

PART A. At which of the times do the two cars pass each other?

Two objects can pass each other only if they have the same position at the same time.

PART C. At which of the lettered times, if any, does car #1 momentarily stop?

The slope on a position versus time graph is the “rise” (change in position) over the “run” (change in time). In physics, the ratio of change in position over change in time is defined as the velocity. Thus, the slope on a position versus time graph is the velocity of the object being graphed.

Solution:

Part A

The slope of the line from 0 to 20 is the constant velocity

$\displaystyle v=\frac{50-25}{20-0}=\frac{25}{20}=1.3\:m/s$

Part B

The line is horizontal at this instant.

$\displaystyle v=0$

Part C

$\displaystyle v=\frac{0-50}{40-30}=\frac{-50}{10}=-5\:m/s$

Arizona State University| PHY 121: Univ Physics I: Mechanics|Homework 1-2| Velocity from Graphs of Position versus Time

PART A. During which trial or trials is the object’s velocity not constant?

The graph of the motion during Trial B has a changing slope and therefore is not constant. The other trials all have graphs with constant slope and thus correspond to motion with constant velocity.

PART B. During which trial or trials is the magnitude of the average velocity the largest?

While Trial B and Trial D do not have the same average velocity, the only difference is the direction! The magnitudes are the same. Neither one is “larger” than the other, and it is only because of how we chose our axes that Trial B has a positive average velocity while Trial D has a negative average velocity. In Trial C the object does not move, so it has an average velocity of zero. During Trial A the object has a positive average velocity but its magnitude is less than that in Trial B and Trial D.

Arizona State University| PHY 121: Univ Physics I: Mechanics|Homework 1-2| Average Velocity from a Position vs. Time Graph

Learning Goal: To learn to read a graph of position versus time and to calculate average velocity.

In this problem, you will determine the average velocity of a moving object from the graph of its position x(t) as a function of time t. A traveling object might move at different speeds and in different directions during an interval of time, but if we ask at what constant velocity the object would have to travel to achieve the same displacement over the given time interval, that is what we call the object’s average velocity. We will use the notation vave[t1,t2] to indicate average velocity over the time interval from t1 to t2. For instance, vave[1,3] is the average velocity over the time interval from t=1 to t=3.

PART A. Consulting the graph shown in the figure, find the object’s average velocity over the time interval from 0 to 1 second.

Average velocity is defined as the constant velocity at which an object would have to travel to achieve a given displacement (difference between final and initial positions, which can be negative) over a given time interval, from the initial time ti to the final time tf. The average velocity is therefore equal to the displacement divided by the given time interval. In symbolic form, average velocity is given by

$v_{ave}\left[t_i,\:t_f\right]=\frac{x\left(t_f\right)-x\left(t_i\right)}{t_f-t_i}$

Since the final and initial positions are equal, the average velocity is 0 m/s.

PART B. Find the average velocity over the time interval from 1 to 3 seconds.

$v_{ave}\left[t_i,\:t_f\right]=\frac{x\left(t_f\right)-x\left(t_i\right)}{t_f-t_i}=\frac{60-20}{3-1}=\frac{40}{2}=20\:m/s$

PART C. Now find $v_{ave}\left[0,\:3\right]$.

$v_{ave}\left[0,\:3\right]=\frac{x\left(t_f\right)-x\left(t_i\right)}{t_f-t_i}=\frac{60-20}{3-0}=\frac{40}{3}=13.3\:m/s$

PART D. Find the average velocity over the time interval from 3 to 6 seconds.

$v_{ave}\left[3,\:6\right]=\frac{x\left(t_f\right)-x\left(t_i\right)}{t_f-t_i}=\frac{20-60}{6-3}=\frac{-40}{3}=-13.3\:m/s$