Path of an Object When an object is at the point and has a velocity vector It moves with

Finding Velocity and Acceleration Along a Plane Curve In Exercises 1-8,the position vector describes the path of an object moving in the xy-plane. (a) Find the velocity vector,speed,and acceleration vector of the object. (b) Evaluate the velocity vector and acceleration vector of the object at the given point. (c) Sketch a graph of the path,and sketch the velocity and acceleration vectors at the given point. Position Vector Point \(\mathbf{r}(t)=3 t \mathbf{i}+(t-1) \mathbf{j}\) (3, 0) Text Transcription: r(t)=3t i + (t-1)j

Finding Velocity and Acceleration Along a Plane Curve In Exercises 1-8,the position vector describes the path of an object moving in the xy-plane. (a) Find the velocity vector,speed,and acceleration vector of the object. (b) Evaluate the velocity vector and acceleration vector of the object at the given point. (c) Sketch a graph of the path,and sketch the velocity and acceleration vectors at the given point. Position Vector Point \(\mathbf{r}(t)=t \mathbf{i}+\left(-t^{2}+4\right) \mathbf{j}\) (1, 3) Text Transcription: r(t)=t i + (-t^2+4)

Finding Velocity and Acceleration Along a Plane Curve In Exercises 1-8,the position vector describes the path of an object moving in the xy-plane. (a) Find the velocity vector,speed,and acceleration vector of the object. (b) Evaluate the velocity vector and acceleration vector of the object at the given point. (c) Sketch a graph of the path,and sketch the velocity and acceleration vectors at the given point. Position Vector Point \(\mathbf{r}(t)=t^{2} \mathbf{i}+t \mathbf{j}\) (4, 2) Text Transcription: r(t)=t^2 i + tj

Finding Velocity and Acceleration Along a Plane Curve In Exercises 1-8,the position vector describes the path of an object moving in the xy-plane. (a) Find the velocity vector,speed,and acceleration vector of the object. (b) Evaluate the velocity vector and acceleration vector of the object at the given point. (c) Sketch a graph of the path,and sketch the velocity and acceleration vectors at the given point. Position Vector Point \(\mathbf{r}(t)=\left(\frac{1}{4} t^{3}+1\right) \mathbf{i}+t \mathbf{j}\) (3, 2) Text Transcription: r(t)=(1/4t^3 + 1)i + tj

Finding Velocity and Acceleration Along a Plane Curve In Exercises 1-8,the position vector describes the path of an object moving in the xy-plane. (a) Find the velocity vector,speed,and acceleration vector of the object. (b) Evaluate the velocity vector and acceleration vector of the object at the given point. (c) Sketch a graph of the path,and sketch the velocity and acceleration vectors at the given point. Position Vector Point \(\mathbf{r}(t)=2 \cos t \mathbf{i}+2 \sin t \mathbf{j}\) (\(\sqrt{2}, \sqrt{2}\)) Text Transcription: r(t)=2 cos t i + 2 sin tj

Finding Velocity and Acceleration Along a Plane Curve In Exercises 1-8,the position vector describes the path of an object moving in the xy-plane. (a) Find the velocity vector,speed,and acceleration vector of the object. (b) Evaluate the velocity vector and acceleration vector of the object at the given point. (c) Sketch a graph of the path,and sketch the velocity and acceleration vectors at the given point. Position Vector Point \(\mathbf{r}(t)=3 \cos t \mathbf{i}+2 \sin t \mathbf{j}\) (3, 0) Text Transcription: r(t)=3 cos t i + 2 sin tj

Finding Velocity and Acceleration Along a Plane Curve In Exercises 1-8,the position vector describes the path of an object moving in the xy-plane. (a) Find the velocity vector,speed,and acceleration vector of the object. (b) Evaluate the velocity vector and acceleration vector of the object at the given point. (c) Sketch a graph of the path,and sketch the velocity and acceleration vectors at the given point. Position Vector Point \(\mathbf{r}(t)=\langle t-\sin t, 1-\cos t\rangle\) (\(\pi, 2\)) Text Transcription: r(t)=<t - sin t, 1 - cos t>

Finding Velocity and Acceleration Along a Plane Curve In Exercises 1-8,the position vector describes the path of an object moving in the xy-plane. (a) Find the velocity vector,speed,and acceleration vector of the object. (b) Evaluate the velocity vector and acceleration vector of the object at the given point. (c) Sketch a graph of the path,and sketch the velocity and acceleration vectors at the given point. Position Vector Point \(\mathbf{r}(t)=\left\langle e^{-t}, e^{t}\right\rangle\) (1, 1) Text Transcription: r(t)=<e^-t, e^t>

Finding Velocity and Acceleration Vectors In Exercises 9-18,the position vector r describes the path of an object moving in space. (a)Find the velocity vector,speed,and acceleration vector of the object. (b)Evaluate the velocity vector and acceleration vector of the object at the given value of t. Position Vector Time \(\mathbf{r}(t)=t \mathbf{i}+5 t \mathbf{j}+3 t \mathbf{k}\) t = 1 Text Transcription: r(t)=ti + 5tj + 3tk

Finding Velocity and Acceleration Vectors In Exercises 9-18,the position vector r describes the path of an object moving in space. (a)Find the velocity vector,speed,and acceleration vector of the object. (b)Evaluate the velocity vector and acceleration vector of the object at the given value of t. Position Vector Time \(\mathbf{r}(t)=4 t \mathbf{i}+4 t \mathbf{j}+2 t \mathbf{k}\) t = 3 Text Transcription: r(t)=4ti + 4tj + 2tk

Finding Velocity and Acceleration Vectors In Exercises 9-18,the position vector r describes the path of an object moving in space. (a)Find the velocity vector,speed,and acceleration vector of the object. (b)Evaluate the velocity vector and acceleration vector of the object at the given value of t. Position Vector Time \(\mathbf{r}(t)=t \mathbf{i}+t^{2} \mathbf{j}+\frac{1}{2} t^{2} \mathbf{k}\) t = 4 Text Transcription: r(t)=ti + t^2j + 1/2t^2k

Finding Velocity and Acceleration Vectors In Exercises 9-18,the position vector r describes the path of an object moving in space. (a)Find the velocity vector,speed,and acceleration vector of the object. (b)Evaluate the velocity vector and acceleration vector of the object at the given value of t. Position Vector Time \(\mathbf{r}(t)=3 t \mathbf{i}+t \mathbf{j}+\frac{1}{4} t^{2} \mathbf{k}\) t = 2 Text Transcription: r(t)=3ti + tj + 1/4t^2k

Finding Velocity and Acceleration Vectors In Exercises 9-18,the position vector r describes the path of an object moving in space. (a)Find the velocity vector,speed,and acceleration vector of the object. (b)Evaluate the velocity vector and acceleration vector of the object at the given value of t. Position Vector Time \(\mathbf{r}(t)=t \mathbf{i}+t \mathbf{j}+\sqrt{9-t^{2}} \mathbf{k}\) t = 0 Text Transcription: r(t)=ti + tj + sqrt 9 - t^2 k

Finding Velocity and Acceleration Vectors In Exercises 9-18,the position vector r describes the path of an object moving in space. (a)Find the velocity vector,speed,and acceleration vector of the object. (b)Evaluate the velocity vector and acceleration vector of the object at the given value of t. Position Vector Time \(\mathbf{r}(t)=t^{2} \mathbf{i}+t \mathbf{j}+2 t^{3 / 2} \mathbf{k}\) t = 4 Text Transcription: r(t)=t^2 i + tj + 2t^3/2 k

Finding Velocity and Acceleration Vectors In Exercises 9-18,the position vector r describes the path of an object moving in space. (a)Find the velocity vector,speed,and acceleration vector of the object. (b)Evaluate the velocity vector and acceleration vector of the object at the given value of t. Position Vector Time \(\mathbf{r}(t)=\langle 4 t, 3 \cos t, 3 \sin t\rangle\) \(t=\pi\) Text Transcription: r(t)=<4t, 3 cos t, 3 sin t> t = pi

Finding Velocity and Acceleration Vectors In Exercises 9-18,the position vector r describes the path of an object moving in space. (a)Find the velocity vector,speed,and acceleration vector of the object. (b)Evaluate the velocity vector and acceleration vector of the object at the given value of t. Position Vector Time \(\mathbf{r}(t)=\left\langle 2 \cos t, 2 \sin t, t^{2}\right\rangle\) \(t=\frac{\pi}{4}\) Text Transcription: r(t)=<2 cos t, 2 sin t, t^2> t = pi/4

Finding Velocity and Acceleration Vectors In Exercises 9-18,the position vector r describes the path of an object moving in space. (a)Find the velocity vector,speed,and acceleration vector of the object. (b)Evaluate the velocity vector and acceleration vector of the object at the given value of t. Position Vector Time \(\mathbf{r}(t)=\left\langle e^{t} \cos t, e^{t} \sin t, e^{t}\right\rangle\) t = 0 Text Transcription: r(t)=<e^t cos t, e^t sin t, e^t>

Finding Velocity and Acceleration Vectors In Exercises 9-18,the position vector r describes the path of an object moving in space. (a)Find the velocity vector,speed,and acceleration vector of the object. (b)Evaluate the velocity vector and acceleration vector of the object at the given value of t. Position Vector Time \(\mathbf{r}(t)=\left\langle\ln t, \frac{1}{t}, t^{4}\right\rangle\) t = 2 Text Transcription: r(t)=<ln t, 1/t, t^4>

Finding a Position Vector by Integration In Exercises 19-24,use the given acceleration vector to find the velocity and position vectors. Then find the position at time t = 2. \(\mathbf{a}(t)=\mathbf{i}+\mathbf{j}+\mathbf{k}, \quad \mathbf{v}(0)=\mathbf{0}, \quad \mathbf{r}(0)=\mathbf{0}\) Text Transcription: a(t) = i +j +k, v(0) =0. r(0) = 0

Finding a Position Vector by Integration In Exercises 19-24,use the given acceleration vector to find the velocity and position vectors. Then find the position at time t = 2. \(\mathbf{a}(t)=2 \mathbf{i}+3 \mathbf{k}, \quad \mathbf{v}(0)=4 \mathbf{j}, \quad \mathbf{r}(0)=\mathbf{0}\) Text Transcription: a(t) 2i+ 3k, v(0) = 4j. r(0) = 0

Finding a Position Vector by Integration In Exercises 19-24,use the given acceleration vector to find the velocity and position vectors. Then find the position at time t = 2. \(\mathbf{a}(t)=t \mathbf{j}+t \mathbf{k}, \quad \mathbf{v}(1)=5 \mathbf{j}, \quad \mathbf{r}(1)=\mathbf{0}\) Text Transcription: a(t) = tj + tk, v(1) 5j. r(1) = 0

Finding a Position Vector by Integration In Exercises 19-24,use the given acceleration vector to find the velocity and position vectors. Then find the position at time t = 2. \(\mathbf{a}(t)=-32 \mathbf{k}, \quad \mathbf{v}(0)=3 \mathbf{i}-2 \mathbf{j}+\mathbf{k}, \quad \mathbf{r}(0)=5 \mathbf{j}+2 \mathbf{k}\) Text Transcription: a(t)=-32k, v(0) = 3i- 2j +k, r(0) 5j + 2k

Finding a Position Vector by Integration In Exercises 19-24,use the given acceleration vector to find the velocity and position vectors. Then find the position at time t = 2. \(\mathbf{a}(t)=-\cos t \mathbf{i}-\sin t \mathbf{j}, \quad \mathbf{v}(0)=\mathbf{j}+\mathbf{k}, \quad \mathbf{r}(0)=\mathbf{i}\) Text Transcription: a(t)=-cos ti - sin tj. v(0) = j + k, r(0) = i

Finding a Position Vector by Integration In Exercises 19-24,use the given acceleration vector to find the velocity and position vectors. Then find the position at time t = 2. \(\mathbf{a}(t)=e^{t} \mathbf{i}-8 \mathbf{k}, \quad \mathbf{v}(0)=2 \mathbf{i}+3 \mathbf{j}+\mathbf{k}, \quad \mathbf{r}(0)=\mathbf{0}\) Text Transcription: a(t) = e^t - 8k, v{0) = 2i +3j + k, r(0) = 0

Projectile Motion In Exercises 25-38,use the model for projectile motion,assuming there is no air resistance. A baseball is hit from a height of 2.5 feet above the ground with an initial velocity of 140 feet per second and at an angle of \(22^{\circ}\) above the horizontal. Find the maximum height reached by the baseball. Determine whether it will clear a 10-foot-high fence located 375 feet from home plate. Text Transcription: 22 degrees

Projectile Motion In Exercises 25-38,use the model for projectile motion,assuming there is no air resistance. Determine the maximum height and range of a projectile fired at a height of 3 feet above the ground with an initial velocity of 900 feet per second and at an angle of \(45^{\circ}\) above the horizontal. Text Transcription: 45 degrees

Projectile Motion In Exercises 25-38,use the model for projectile motion,assuming there is no air resistance. A baseball,hit 3 feet above the ground,leaves the bat at an angle of \(45^{\circ}\) and is caught by an outfielder 3 feet above the ground and 300 feet from home plate. What is the initial speed of the ball,and how high does it rise? Text Transcription: 45 degrees

Projectile Motion In Exercises 25-38,use the model for projectile motion,assuming there is no air resistance. A baseball player at second base throws a ball 90 feet to the player at first base. The ball is released at a point 5 feet above the ground with an initial velocity of 50 miles per hour and at an angle of \(15^{\circ}\) above the horizontal. At what height does the player at first base catch the ball? Text Transcription: 15 degrees

Projectile Motion In Exercises 25-38,use the model for projectile motion,assuming there is no air resistance. Eliminate the parameter t from the position vector for the motion of a projectile to show that the rectangular equation is \(y=-\frac{16 \sec ^{2} \theta}{v_{0}^{2}} x^{2}+(\tan \theta) x+h\). Text Transcription: y=-16 sec^2 theta/v_0^2 x^2+(tan theta)x+h

Projectile Motion In Exercises 25-38,use the model for projectile motion,assuming there is no air resistance. The path of a ball is given by the rectangular equation \(y=x-0.005 x^{2}\) Text Transcription: y=x-0.005 x^2

Projectile Motion In Exercises 25-38,use the model for projectile motion,assuming there is no air resistance. The Rogers Centre in Toronto, Ontario, has a center field fence that is 10 1feet high and 400 feet from home plate. A ball is hit 3 feet above the ground and leaves the bat at a speed of 100 miles per hour. (a) The ball leaves the bat at an angle of \(\theta=\theta_{0}\) with the horizontal. Write the vector-valued function for the path of the ball. (b) Use a graphing utility to graph the vector-valued function for \(\theta_{0}=10^{\circ}, \theta_{0}=15^{\circ}, \theta_{0}=20^{\circ}, \text { and } \theta_{0}=25^{\circ}\). Use the graphs to approximate the minimum angle required for the hit to be a home run. (c) Determine analytically the minimum angle required for the hit to be a home run. Text Transcription: theta=theta_0 theta_0=10 degrees, theta_0=15 degrees, theta_0=20, and theta_0=25 degrees

Projectile Motion In Exercises 25-38,use the model for projectile motion,assuming there is no air resistance. Football The quarterback of a football team releases a pass at a height of 7 feet above the playing field, and the football is caught by a receiver 30 yards directly downfield at a height of 4 feet. The pass is released at an angle of \(35^{\circ}\) with the horizontal. (a) Find the speed of the football when it is released. (b) Find the maximum height of the football. (c) Find the time the receiver has to reach the proper position after the quarterback releases the football. Text Transcription: 35 degrees

Projectile Motion In Exercises 25-38,use the model for projectile motion,assuming there is no air resistance. A bale ejector consists of two variable-speed belts at the end of a baler. Its purpose is to toss bales into a trailing wagon. In loading the back of a wagon,a bale must be thrown to a position 8 feet above and 16 feet behind the ejector. (a)Find the minimum initial speed of the bale and the corresponding angle at which it must be ejected from the baler. (b)The ejector has a fixed angle of \(45^{\circ}\) Find the initial speed required. Text Transcription: 45 degrees

Projectile Motion In Exercises 25-38,use the model for projectile motion,assuming there is no air resistance. A bomber is flying at an altitude of 30,000 feet at a speed of 540 miles per hour (see figure). When should the bomb be released for it to hit the target? (Give your answer in terms of the angle of depression from the plane to the target.) What is the speed of the bomb at the time of impact?

Projectile Motion In Exercises 25-38,use the model for projectile motion,assuming there is no air resistance. A shot fired from a gun with a muzzle velocity of 1200 feet per second is to hit a target 3000 feet away. Determine the minimum angle of elevation of the gun.

Projectile Motion In Exercises 25-38,use the model for projectile motion,assuming there is no air resistance. A projectile is fired from ground level at an angle of \(12^{\circ}\) with the horizontal. The projectile is to have a range of 200 feet. Find the minimum initial velocity necessary. Text Transcription: 12 degrees

Projectile Motion In Exercises 25-38,use the model for projectile motion,assuming there is no air resistance. Use a graphing utility to graph the paths of a projectile for the given values of \(\theta\) and \(v_{0}\). For each case, use the graph to approximate the maximum height and range of the projectile. (Assume that the projectile is launched from ground level.) (a) \(\theta=10^{\circ}, v_{0}=66 \mathrm{ft} / \mathrm{sec}\) (b) \(\theta=10^{\circ}, v_{0}=146 \mathrm{ft} / \mathrm{sec}\) (c) \(\theta=45^{\circ}, v_{0}=66 \mathrm{ft} / \mathrm{sec}\) (d) \(\theta=45^{\circ}, v_{0}=146 \mathrm{ft} / \mathrm{sec}\) (e) \(\theta=60^{\circ}, v_{0}=66 \mathrm{ft} / \mathrm{sec}\) (f) \(\theta=60^{\circ}, v_{0}=146 \mathrm{ft} / \mathrm{sec}\) Text Transcription: theta=10 degrees, v_0 = 66ft/sec theta=10 degrees, v_0 = 146ft/sec theta=45 degrees, v_0 = 66ft/sec theta=45 degrees, v_0 = 146ft/sec theta=60 degrees, v_0 = 66ft/sec theta=60 degrees, v_0 = 146ft/sec

Projectile Motion In Exercises 25-38,use the model for projectile motion,assuming there is no air resistance. Find the angles at which an object must be thrown to obtain (a) the maximum range and (b) the maximum height.

Projectile Motion In Exercises 39 and 40, use the model for projectile motion, assuming there is no air resistance. [g = -9.8 meters per second per second] Determine the maximum height and range of a projectile fired at a height of 1.5 meters above the ground with an initial velocity of 100 meters per second and at an angle of \(30^{\circ}\) above the horizontal. Text Transcription: 30 degrees

Projectile Motion In Exercises 39 and 40, use the model for projectile motion, assuming there is no air resistance. [g = -9.8 meters per second per second] A projectile is fired from ground level at an angle of \(8^{\circ}\) with the horizontal. The projectile is to have a range of 50 meters. Find the minimum initial velocity necessary. Text Transcription: 8 degrees

Shot-Put Throw The path of a shot thrown at an angle \(\theta\) is \(\mathbf{r}(t)=\left(v_{0} \cos \theta\right) t \mathbf{i}+\left[h+\left(v_{0} \sin \theta\right) t-\frac{1}{2} g t^{2}\right] \mathbf{j}\) where \(v_{0}\) is the initial speed, h is the initial height, t is the time in seconds, and g is the acceleration due to gravity. Verify that the shot will remain in the air for a total of \(t=\frac{v_{0} \sin \theta+\sqrt{v_{0}^{2} \sin ^{2} \theta+2 g h}}{g}\) seconds and will travel a horizontal distance of \(\frac{v_{0}^{2} \cos \theta}{g}\left(\sin \theta+\sqrt{\sin ^{2} \theta+\frac{2 g h}{v_{0}^{2}}}\right) \text { feet }\) feet. Text Transcription: theta r(t) = (v_0 cos theta)t i + [h + (v_0 sin theta)t - 1/2 gt^2]j v_0 t=v_0 sin theta + sqrt v_0^2 sin^2 theta + 2gh/g v_0^2 cos theta/g (sin theta + sqrt sin^2 theta + 2gh/v_0^2)

Shot-Put Throw A shot is thrown from a height of h = 6 feet with an initial speed of \(v_{0}=45\) feet per second and at an angle of \(\theta=42.5^{\circ}\) with the horizontal. Use the result of Exercise 41 to find the total time of travel and the total horizontal distance traveled. Text Transcription: v_0=45 theta=42.5 degrees

Cycloidal Motion In Exercises 43 and 44, consider the motion of a point (or particle) on the circumference of a rolling circle. As the circle rolls, it generates the cycloid \(r(t)=b(\omega t-\sin \omega t) i+b(1-\cos \omega t) \mathbf{j}\) where \(\omega\) is the constant angular velocity of the circle and b is the radius of the circle. Find the velocity and acceleration vectors of the particle. Use the results to determine the times at which the speed of the particle will be (a) zero and (b) maximized. Text Transcription: r(t)=b(omega t-sin omega t) i+b(1-cos omega t)j omega

Cycloidal Motion In Exercises 43 and 44, consider the motion of a point (or particle) on the circumference of a rolling circle. As the circle rolls, it generates the cycloid \(r(t)=b(\omega t-\sin \omega t) i+b(1-\cos \omega t) \mathbf{j}\) where \(\omega\) is the constant angular velocity of the circle and b is the radius of the circle. Find the maximum speed of a point on the circumference of an automobile tire of radius I foot when the automobile is traveling at 60 miles per hour. Compare this speed with the speed of the automobile. Text Transcription: r(t)=b(omega t-sin omega t) i+b(1-cos omega t)j omega

Circular Motion In Exercises 45-48, consider a particle moving on a circular path of radius b described by \(r(t)=b \cos \omega t \mathrm{i}+b \sin \omega t \mathrm{j}\), where \(\omega=d u / d t\) is the constant angular velocity. Find the velocity vector and show that it is orthogonal to r(t). Text Transcription: r(t)=b cos omega t i + b sin omega t j omega=du/dt

Circular Motion In Exercises 45-48, consider a particle moving on a circular path of radius b described by \(r(t)=b \cos \omega t \mathrm{i}+b \sin \omega t \mathrm{j}\), where \(\omega=d u / d t\) is the constant angular velocity. (a) Show that the speed of the particle is \(b \omega\). (b) Use a graphing utility in parametric mode to graph the circle for b = 6. Try different values of \(\omega\). Does the graphing utility draw the circle faster for greater values of \(\omega\)? Text Transcription: r(t)=b cos omega t i + b sin omega t j omega=du/dt b omega

Circular Motion In Exercises 45-48, consider a particle moving on a circular path of radius b described by \(r(t)=b \cos \omega t \mathrm{i}+b \sin \omega t \mathrm{j}\), where \(\omega=d u / d t\) is the constant angular velocity. Find the acceleration vector and show that its direction is always toward the center of the circle. Text Transcription: r(t)=b cos omega t i + b sin omega t j omega=du/dt

Circular Motion In Exercises 45-48, consider a particle moving on a circular path of radius b described by \(r(t)=b \cos \omega t \mathrm{i}+b \sin \omega t \mathrm{j}\), where \(\omega=d u / d t\) is the constant angular velocity. Show that the magnitude of the acceleration vector is \(b \omega^{2}\). Text Transcription: r(t)=b cos omega t i + b sin omega t j omega=du/dt b omega^2

Circular Motion In Exercises 49 and 50, use the results of Exercises 45-48. A stone weighing 1 pound is attached to a two-foot string and is whirled horizontally (see figure). The string will break under a force of 10 pounds. Find the maximum speed the stone can attain without breaking the string. (Use F = ma, where \(m=\frac{1}{32}\).) Text Transcription: m=1/32

Circular Motion In Exercises 49 and 50, use the results of Exercises 45-48. A 3400-pound automobile is negotiating a circular interchange of radius 300 feet at 30 miles per hour (see figure). Assuming the roadway is level. find the force between the tires and the road such that the car stays on the circular path and does not skid. (Use F = ma, where m = 3400/32.) Find the angle at which the roadway should be banked so that no lateral frictional force is exerted on the tires of the automobile.

Velocity and Speed In your own words, explain the difference between the velocity of an object and its speed.

Particle Motion Consider a particle that is moving on the path \(\mathbf{r}_{1}(t)=x(t) \mathbf{i}+y(t) \mathbf{j}+z(t) \mathbf{k}\). (a) Discuss any changes in the position, velocity, or acceleration of the particle when its position is given by the vector-valued function \(\mathbf{r}_{2}(t)=\mathbf{r}_{1}(2 t)\). (b) Generalize the results for the vector-valued function \(\mathbf{r}_{3}(t)=\mathbf{r}_{1}(\omega t)\). Text Transcription: r_1(t)=x(t)i + y(t)j + z(t) k r_2(t)=r_1(2t) r_3(t)=r_1(omega t)

Proof Prove that when an object is traveling at a constant speed, its velocity and acceleration vectors are orthogonal.

Proof Prove that an object moving in a straight line at a constant speed has an acceleration of 0.

Investigation A particle moves on an elliptical path given by the vector-valued function r(t) = 6 cos ti +3 sin tj. (a) Find v(t), ||v(t)||, and a(t). (b) Use a graphing utility to complete the table. (c) Graph the elliptical path and the velocity and acceleration vectors at the values of t given in the table in part (b). (d) Use the results of parts (b) and (c) to describe the geometric relationship between the velocity and acceleration vectors when the speed of the particle is increasing, and when it is decreasing.

Particle Motion Consider a particle moving on an elliptical path described by \(\mathbf{r}(t)=a \cos \omega t \mathbf{i}+b \sin \omega t \mathbf{j}\). where \(\omega=d \theta / d t\) is the constant angular velocity. (a) Find the velocity vector. What is the speed of the particle? (b) Find the acceleration vector and show that its direction is always toward the center of the ellipse. Text Transcription: r(t)=a cos omega t i + b sin omega t j omega=d theta/dt

Path of an Object When t = 0, an object is at the point (0, 1) and has a velocity vector v(0) = -i. It moves with an acceleration of a(t) = sin ti - cos 1j. Show that the path of the object is a circle.

HOW DO YOU SEE IT? The graph shows the path of a projectile and the velocity and acceleration vectors at times \(t_{1}\) and \(t_{2}\). Classify the angle between the velocity vector and the acceleration vector at times \(t_{1}\) and \(t_{2}\). Is the speed increasing or decreasing at times \(t_{1}\) and \(t_{2}\)? Explain your reasoning. Text Transcription: t_1 t_2

True or False? In Exercises 59-62,determine whether the statement is true or false. If it is false,explain why or give an example that shows it is false. The acceleration of an object is the derivative of the speed.

True or False? In Exercises 59-62,determine whether the statement is true or false. If it is false,explain why or give an example that shows it is false. The velocity of an object is the derivative of the position.

True or False? In Exercises 59-62,determine whether the statement is true or false. If it is false,explain why or give an example that shows it is false. The velocity vector points in the direction of motion.

True or False? In Exercises 59-62,determine whether the statement is true or false. If it is false,explain why or give an example that shows it is false. If a particle moves along a straight line,then the velocity and acceleration vectors are orthogonal.