Galileo’s free-fall formula Galileo developed a formula

Exercises 31–34 give the position function \(s=f(t)\) of an object moving along the \(s\) - axis as a function of time \(t\). Graph \(f\) together with the velocity function \(y(t)=d s / d t=f^{\prime}(t)\) and the acceleration function \(a(t)=d^{2} s / d t^{2}\). Comment on the object’s behavior in relation to the signs and values of \(v\) and \(a\). Include in your commentary such topics as the following: a. When is the object momentarily at rest? b. When does it move to the left (down) or to the right (up)? c. When does it change direction? d. When does it speed up and slow down? e. When is it moving fastest (highest speed)? Slowest? f. When is it farthest from the axis origin? \(s=4-7 t+6 t^{2}-t^{3}, \quad 0 \leq t \leq 4\) Equation Transcription: Text Transcription: s = f(t) s t f y(t) = ds/dt = f’(t) a(t) = d^2 s/dt^2 v a s = 4 - 7t + 6t^2 - t^3, 0 leq t leq 4

Problem 2E Exercises 1–6 give the positions s = f(t) of a body moving on a coordinate line, with s in meters and t in seconds. a. Find the body’s displacement and average velocity for the given time interval. b. Find the body’s speed and acceleration at the endpoints of the interval. c. When, if ever, during the interval does the body change direction?

Problem 1E Exercises 1–6 give the positions s = f(t) of a body moving on a coordinate line, with s in meters and t in seconds. a. Find the body’s displacement and average velocity for the given time interval. b. Find the body’s speed and acceleration at the endpoints of the interval. c. When, if ever, during the interval does the body change direction?

Exercises 1–6 give the positions \(s=f(t)\) of a body moving on a coordinate line, with \(s\) in meters and \(t\) in seconds. a. Find the body’s displacement and average velocity for the given time interval. b. Find the body’s speed and acceleration at the endpoints of the interval. c. When, if ever, during the interval does the body change direction? \(s=-t^{3}+3 t^{2}-3 t, \quad 0 \leq t \leq 3\) Equation Transcription: Text Transcription: s = f(t) s t s = ?t^3 + 3t^2 ? 3t, 0 leq t leq 3

Problem 4E Exercises 1–6 give the positions s = f(t) of a body moving on a coordinate line, with s in meters and t in seconds. a. Find the body’s displacement and average velocity for the given time interval. b. Find the body’s speed and acceleration at the endpoints of the interval. c. When, if ever, during the interval does the body change direction?

Problem 5E Exercises 1–6 give the positions s = f(t) of a body moving on a coordinate line, with s in meters and t in seconds. a. Find the body’s displacement and average velocity for the given time interval. b. Find the body’s speed and acceleration at the endpoints of the interval. c. When, if ever, during the interval does the body change direction?

Exercises 1–6 give the positions \(s=f(t)\) of a body moving on a coordinate line, with \(s\) in meters and \(t\) in seconds. a. Find the body’s displacement and average velocity for the given time interval. b. Find the body’s speed and acceleration at the endpoints of the interval. c. When, if ever, during the interval does the body change direction? \(s=\frac{25}{t+5}, \quad-4 \leq t \leq 0\) Equation Transcription: Text Transcription: s = f(t) s t s = frac{25}{t + 5}, -4 leq t leq 0

Problem 8E Particle motion At time t ? 0, the velocity of a body moving along the horizontal s-axis is v = t2 - 4t + 3. a. Find the body’s acceleration each time the velocity is zero. b. When is the body moving forward? Backward? c. When is the body’s velocity increasing? Decreasing?

Problem 7E Particle motion At time t, the position of a body moving along the s-axis is = t3 - 6t2 + 9t m. a. Find the body’s acceleration each time the velocity is zero. b. Find the body’s speed each time the acceleration is zero. c. Find the total distance traveled by the body from t = 0 to t = 2.

Problem 10E Lunar projectile motion A rock thrown vertically upward from the surface of the moon at a velocity of 24 m/sec (about 86 km/h) reaches a height of s = 24t - 0.8t2 m in t sec. a. Find the rock’s velocity and acceleration at time t. (The acceleration in this case is the acceleration of gravity on the moon.) b. How long does it take the rock to reach its highest point? c. How high does the rock go? d. How long does it take the rock to reach half its maximum height? e. How long is the rock aloft?

Problem 9E Free fall on Mars and Jupiter The equations for free fall at the surfaces of Mars and Jupiter (s in meters, t in seconds) are s = 1.86t2 on Mars and s = 11.44t2 on Jupiter. How long does it take a rock falling from rest to reach a velocity of 27.8 m/sec (about 100 km h) on each planet?

Problem 11E Finding g on a small airless planet Explorers on a small airless planet used a spring gun to launch a ball bearing vertically upward from the surface at a launch velocity of 15 m/sec. Because the acceleration of gravity at the planet’s surface was gSm/sec2, the explorers expected the ball bearing to reach a height of s = 15t – (1/2)gSt2 m t sec later. The ball bearing reached its maximum height 20 sec after being launched. What was the value of gS?

Problem 12E Speeding bullet A 45-caliber bullet shot straight up from the surface of the moon would reach a height of s = 832t - 2.6t2 ft after t sec. On Earth, in the absence of air, its height would be s = 832t - 16t2 ft after t sec. How long will the bullet be aloft in each case? How high will the bullet go?

Problem 13E Free fall from the Tower of Pisa Had Galileo dropped a cannonball from the Tower of Pisa, 179 ft above the ground, the ball’s height above the ground t sec into the fall would have been s = 179 - 16t2. a. What would have been the ball’s velocity, speed, and acceleration at time t ? b. About how long would it have taken the ball to hit the ground? c. What would have been the ball’s velocity at the moment of impact?

Problem 14E Galileo’s free-fall formula Galileo developed a formula for a body’s velocity during free fall by rolling balls from rest down increasingly steep inclined planks and looking for a limiting formula that would predict a ball’s behavior when the plank was vertical and the ball fell freely; see part (a) of the accompanying figure. He found that, for any given angle of the plank, the ball’s velocity t sec into motion was a constant multiple of t. That is, the velocity was given by a formula of the form v = kt. The value of the constant k depended on the inclination of the plank. In modern notation—part (b) of the figure—with distance in meters and time in seconds, what Galileo determined by experiment was that, for any given angle ?, the ball’s velocity t sec into the roll was a. What is the equation for the ball’s velocity during free fall? b. Building on your work in part (a), what constant acceleration does a freely falling body experience near the surface of Earth?

Problem 15E The accompanying figure shows the velocity v = ds/dt = ƒ(t) (m/sec) of a body moving along a coordinate line. a. When does the body reverse direction? b. When (approximately) is the body moving at a constant speed? c. Graph the body’s speed for 0 ? t ? 10. d. Graph the acceleration, where defined.

Problem 16E A particle P moves on the number line shown in part (a) of the accompanying figure. Part (b) shows the position of P as a function of time t. a. When is P moving to the left? Moving to the right? Standing still? b. Graph the particle’s velocity and speed (where defined).

Launching a rocket When a model rocket is launched, the propellant burns for a few seconds, accelerating the rocket upward. After burnout, the rocket coasts upward for a while and then begins to fall. A small explosive charge pops out a parachute shortly after the rocket starts down. The parachute slows the rocket to keep it from breaking when it lands. The figure here shows velocity data from the flight of the model rocket. Use the data to answer the following. a. How fast was the rocket climbing when the engine stopped? b. For how many seconds did the engine burn? c. When did the rocket reach its highest point? What was its velocity then? d. When did the parachute pop out? How fast was the rocket falling then? e. How long did the rocket fall before the parachute opened? f. When was the rocket’s acceleration greatest? g. When was the acceleration constant? What was its value then (to the nearest integer)?

Problem 18E The accompanying figure shows the velocity v = f(t) of a particle moving on a horizontal coordinate line. a. When does the particle move forward? Move backward? Speed up? Slow down? b. When is the particle’s acceleration positive? Negative? Zero? c. When does the particle move at its greatest speed? d. When does the particle stand still for more than an instant?

Two falling balls The multiflash photograph in the accompanying figure shows two balls falling from rest. The vertical rulers are marked in centimeters. Use the equation s = 490t2 (the freefall equation for \(s\) in centimeters and \(t\) in seconds) to answer the following questions. a. How long did it take the balls to fall the first 160 cm? What was their average velocity for the period? b. How fast were the balls falling when they reached the 160-cm mark? What was their acceleration then? c. About how fast was the light flashing (flashes per second)? Equation Transcription: Text Transcription: s = 490^2 s t

Problem 20E A traveling truck The accompanying graph shows the position s of a truck traveling on a highway. The truck starts at t = 0 and returns 15 h later at t = 15. a. Use the technique described in Section 3.2, Example 3, to graph the truck’s velocity v = ds/dt for 0 ? t ? 15. Then repeat the process, with the velocity curve, to graph the truck’s acceleration dv/dt. b. Suppose that s = 15t2 - t3. Graph ds/dt and d2s/dt2 and compare your graphs with those in part (a).

The graphs in the accompanying figure show the position \(s\), velocity \(v=d s / d t\), and acceleration \(a=d^{2} s / d t\) of a body moving along a coordinate line as functions of time \(t\). Which graph is which? Give reasons for your answers. Equation Transcription: Text Transcription: s v = ds/dt a = d^2 s/dt t

Problem 23E Marginal cost Suppose that the dollar cost of producing x washing machines is c(x) = 2000 + 100x - 0.1x2. a. Find the average cost per machine of producing the first 100 washing machines. b. Find the marginal cost when 100 washing machines are produced. c. Show that the marginal cost when 100 washing machines are produced is approximately the cost of producing one more washing machine after the first 100 have been made, by calculating the latter cost directly.

Problem 22E The graphs in the accompanying figure show the position s, the velocity v = ds/dt, and the acceleration a = d2s/dt2 of a body moving along the coordinate line as functions of time t. Which graph is which? Give reasons for your answers.

Problem 24E Marginal revenue Suppose that the revenue from selling x washing machines is dollars. a. Find the marginal revenue when 100 machines are produced. b. Use the function r’(x) to estimate the increase in revenue that will result from increasing production from 100 machines a week to 101 machines a week. c. Find the limit of r’(x) as How would you interpret this number?

Problem 25E Bacterium population When a bactericide was added to a nutrient broth in which bacteria were growing, the bacterium population continued to grow for a while, but then stopped growing and began to decline. The size of the population at time t (hours) was b = 106 + 104t - 103t2. Find the growth rates at a. t = 0 hours . b. t = 5 hours . c. t = 10 hours .

Problem 26E Draining a tank The number of gallons of water in a tank t minutes after the tank has started to drain is Q(t) = 200(30 – t)2. How fast is the water running out at the end of 10 min? What is the average rate at which the water flows out during the first 10 min?

Problem 27E Draining a tank It takes 12 hours to drain a storage tank by opening the valve at the bottom. The depth y of fluid in the tank t hours after the valve is opened is given by the formula a. Find the rate dy/dt(m/h) at which the tank is draining at time t. b. When is the fluid level in the tank falling fastest? Slowest? What are the values of dy/dt at these times? c. Graph y and dy/dt together and discuss the behavior of y in relation to the signs and values of dt/dt.

Problem 28E Inflating a balloon The volume V = (4/3)?r3 of a spherical balloon changes with the radius. a. At what rate (ft3/ft) does the volume change with respect to the radius when r = 2 ft? b. By approximately how much does the volume increase when the radius changes from 2 to 2.2 ft?

Problem 30E Volcanic lava fountains Although the November 1959 Kilauea Iki eruption on the island of Hawaii began with a line of fountains along the wall of the crater, activity was later confined to a single vent in the crater’s floor, which at one point shot lava 1900 ft straight into the air (a Hawaiian record). What was the lava’s exit velocity in feet per second? In miles per hour? (Hint: If v0 is the exit velocity of a particle of lava, its height t sec later will be s = v0t - 16t2 ft. Begin by finding the time at which ds/dt = 0. Neglect air resistance.)

Problem 29E Airplane takeoff Suppose that the distance an aircraft travels along a runway before takeoff is given by D = (10/9)t2, where D is measured in meters from the starting point and t is measured in seconds from the time the brakes are released. The aircraft will become airborne when its speed reaches 200 km/h. How long will it take to become airborne, and what distance will it travel in that time?

Problem 31E Exercises 31–34 give the position function s = ƒ(t) of an object moving along the s-axis as a function of time t. Graph ƒ together with the velocity function and the acceleration function velocity function v(t) = ds/dt = ƒ’(t) and the acceleration function a(t) = d2s/dt2 = ƒ’’(t). Comment on the object’s behavior in relation to the signs and values of v and a. Include in your commentary such topics as the following: a. When is the object momentarily at rest? b. When does it move to the left (down) or to the right (up)? c. When does it change direction? d. When does it speed up and slow down? e. When is it moving fastest (highest speed)? Slowest? f. When is it farthest from the axis origin? s = 200t - 16t2, 0 ? t ? 12.5 (a heavy object fired straight up from Earth’s surface at 200 ft/sec)

Exercises 31–34 give the position function \(s=f(t)\) of an object moving along the \(s\) - axis as a function of time \(t\). Graph \(f\) together with the velocity function \(y(t)=d s / d t=f^{\prime}(t)\) and the acceleration function \(a(t)=d^{2} s / d t^{2}\). Comment on the object’s behavior in relation to the signs and values of \(v\) and \(a\). Include in your commentary such topics as the following: a. When is the object momentarily at rest? b. When does it move to the left (down) or to the right (up)? c. When does it change direction? d. When does it speed up and slow down? e. When is it moving fastest (highest speed)? Slowest? f. When is it farthest from the axis origin? \(s=t^{2}-3 t+2, \quad 0 \leq t \leq 5\) Equation Transcription: Text Transcription: s = f(t) s t f y(t) = ds/dt = f’(t) a(t) = d^2 s/dt^2 v a s = t^2 - 3t + 2, 0 leq t leq 5

Problem 33E Exercises 31–34 give the position function s = ƒ(t) of an object moving along the s-axis as a function of time t. Graph ƒ together with the velocity function and the acceleration function velocity function v(t) = ds/dt = ƒ’(t) and the acceleration function a(t) = d2s/dt2 = ƒ’’(t). Comment on the object’s behavior in relation to the signs and values of v and a. Include in your commentary such topics as the following: a. When is the object momentarily at rest? b. When does it move to the left (down) or to the right (up)? c. When does it change direction? d. When does it speed up and slow down? e. When is it moving fastest (highest speed)? Slowest? f. When is it farthest from the axis origin?