Cycloidal Motion In Exercises 47 and 48, consider the motion of a point (or particle) on

In Exercises 1–10, the position vector r describes the path of an object moving in the xy-plane. Sketch a graph of the path and sketch the velocity and acceleration vectors at the given point. Position Function Point \(\mathbf{r}(t)=3 t \mathbf{i}+(t-1) \mathbf{j}\) (3,0) Text Transcription: r(t)=3 t i+(t-1) j

In Exercises 1–10, the position vector r describes the path of an object moving in the xy-plane. Sketch a graph of the path and sketch the velocity and acceleration vectors at the given point. Position Function Point \(\mathbf{r}(t)=(6-t) \mathbf{i}+t \mathbf{j}\) (3,3) Text Transcription: r(t)=(6-t) i+t j

In Exercises 1–10, the position vector r describes the path of an object moving in the xy-plane. Sketch a graph of the path and sketch the velocity and acceleration vectors at the given point. Position Function Point \(\mathbf{r}(t)=t^{2} \mathbf{i}+t \mathbf{j}\) (4,2) Text Transcription: r(t)=t^2 i+t j

In Exercises 1–10, the position vector r describes the path of an object moving in the xy-plane. Sketch a graph of the path and sketch the velocity and acceleration vectors at the given point. Position Function 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) j

In Exercises 1–10, the position vector r describes the path of an object moving in the xy-plane. Sketch a graph of the path and sketch the velocity and acceleration vectors at the given point. Position Function Point \(\mathbf{r}(t)=t^{2} \mathbf{i}+t^{3} \mathbf{j}\) (1,1) Text Transcription: r(t)=t^2 i+t^3 j

In Exercises 1–10, the position vector r describes the path of an object moving in the xy-plane. Sketch a graph of the path and sketch the velocity and acceleration vectors at the given point. Position Function Point \(\mathbf{r}(t)=\left(\frac{1}{4} t^{3}+1\right) \mathbf{i}+t \mathbf{j}\) (3,2) Text Transcription: r(t)=(1/4 t^3+1) i+t j

In Exercises 1–10, the position vector r describes the path of an object moving in the xy-plane. Sketch a graph of the path and sketch the velocity and acceleration vectors at the given point. Position Function 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 t j (sqrt{2}, sqrt{2})

In Exercises 1–10, the position vector r describes the path of an object moving in the xy-plane. Sketch a graph of the path and sketch the velocity and acceleration vectors at the given point. Position Function 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 t j

In Exercises 1–10, the position vector r describes the path of an object moving in the xy-plane. Sketch a graph of the path and sketch the velocity and acceleration vectors at the given point. Position Function Point \(\mathbf{r}(t)=\langle t-\sin t, 1-\cos t\rangle\) (\(\pi\), 2) Text Transcription: r(t)=langle t-sin t, 1-cos t rangle pi

In Exercises 1–10, the position vector r describes the path of an object moving in the xy-plane. Sketch a graph of the path and sketch the velocity and acceleration vectors at the given point. Position Function Point \(\mathbf{r}(t)=\left\langle e^{-t}, e^{t}\right\rangle\) (1,1) Text Transcription: r(t)=langle e^-t, e^t rangle

In Exercises 11–20, the position vector r describes the path of an object moving in space. Find the velocity, speed, and acceleration of the object. \(\mathbf{r}(t)=t \mathbf{i}+5 t \mathbf{j}+3 t \mathbf{k}\) Text Transcription: r(t)=t i+5 t j+3 t k

In Exercises 11–20, the position vector r describes the path of an object moving in space. Find the velocity, speed, and acceleration of the object. \(\mathbf{r}(t)=4 t \mathbf{i}+4 t \mathbf{j}+2 t \mathbf{k}\) Text Transcription: r(t)=4 t i+4 t j+2 t k

In Exercises 11–20, the position vector r describes the path of an object moving in space. Find the velocity, speed, and acceleration of the object. \(\mathbf{r}(t)=t \mathbf{i}+t^{2} \mathbf{j}+\frac{t^{2}}{2} \mathbf{k}\) Text Transcription: r(t)=t i+t^2 j+ t^2/2 k

In Exercises 11–20, the position vector r describes the path of an object moving in space. Find the velocity, speed, and acceleration of the object. \(\mathbf{r}(t)=3 t \mathbf{i}+t \mathbf{j}+\frac{1}{4} t^{2} \mathbf{k}\) Text Transcription: r(t)=3 t i+t j+ 1/4 t^2 k

In Exercises 11–20, the position vector r describes the path of an object moving in space. Find the velocity, speed, and acceleration of the object. \(\mathbf{r}(t)=t \mathbf{i}+t \mathbf{j}+\sqrt{9-t^{2}} \mathbf{k}\) Text Transcription: r(t)=t i+t j+ sqrt9-t^2 k

In Exercises 11–20, the position vector r describes the path of an object moving in space. Find the velocity, speed, and acceleration of the object. \(\mathbf{r}(t)=t^{2} \mathbf{i}+t \mathbf{j}+2 t^{3 / 2} \mathbf{k}\) Text Transcription: r(t)=t^2 i+t j+2 t^3 / 2 k

In Exercises 11–20, the position vector r describes the path of an object moving in space. Find the velocity, speed, and acceleration of the object. \(\mathbf{r}(t)=\langle 4 t, 3 \cos t, 3 \sin t\rangle\) Text Transcription: r(t)= langle 4 t, 3 cos t, 3 sin t rangle

In Exercises 11–20, the position vector r describes the path of an object moving in space. Find the velocity, speed, and acceleration of the object. \(\mathbf{r}(t)=\left\langle 2 \cos t, 2 \sin t, t^{2}\right\rangle\) Text Transcription: r(t)= langle 2 cos t, 2 sin t, t^2 rangle

In Exercises 11–20, the position vector r describes the path of an object moving in space. Find the velocity, speed, and acceleration of the object. \(\mathbf{r}(t)=\left\langle e^{t} \cos t, e^{t} \sin t, e^{t}\right\rangle\) Text Transcription: r(t)= langle e^t cos t, e^t sin t, e^t rangle

In Exercises 11–20, the position vector r describes the path of an object moving in space. Find the velocity, speed, and acceleration of the object. \(\mathbf{r}(t)=\left\langle\ln t, \frac{1}{t}, t^{4}\right\rangle\) Text Transcription: r(t)= langle ln t, 1/t, t^4 rangle

Linear Approximation In Exercises 21 and 22, the graph of the vector-valued function r(t) and a tangent vector to the graph at t=\(t_{0}\) are given. (a) Find a set of parametric equations for the tangent line to the graph at t=\(t_{0}\) (b) Use the equations for the line to approximate r(\(t_{0}\) + 0.1) \(\mathbf{r}(t)=\left\langle t,-t^{2}, \frac{1}{4} t^{3}\right\rangle\), \(t_{0}\)=1 Text Transcription: r(t)= langle t,-t^2, 1/4 t^3 rangle t_0

Linear Approximation In Exercises 21 and 22, the graph of the vector-valued function r(t) and a tangent vector to the graph at t=\(t_{0}\) are given. (a) Find a set of parametric equations for the tangent line to the graph at t=\(t_{0}\) (b) Use the equations for the line to approximate r(\(t_{0}\) + 0.1) \(\mathbf{r}(t)=\left\langle t, \sqrt{25-t^{2}}, \sqrt{25-t^{2}}\right\rangle\), \(t_{0}\)=3 Text Transcription: r(t)=langle t, sqrt25-t^2, sqrt25-t^2 rangle t_0

In Exercises 23–28, use the given acceleration function to find the velocity and position vectors. Then find the position at time t=2. \(\mathbf{a}(t)=\mathbf{i}+\mathbf{j}+\mathbf{k}\) v(0)=0, r(0)=0 Text Transcription: a(t)=i+j+k

In Exercises 23–28, use the given acceleration function to find the velocity and position vectors. Then find the position at time t=2. \(\mathbf{a}(t)=2 \mathbf{i}+3 \mathbf{k} \) \(\mathbf{v}(0)=4 \mathbf{j}\), \(\mathbf{r}(0)=\mathbf{0}\) Text Transcription: a(t)=2 i+3 k v(0)=4 j r(0)=0

In Exercises 23–28, use the given acceleration function to find the velocity and position vectors. Then find the position at time t=2. \(\mathbf{a}(t)=t \mathbf{j}+t \mathbf{k} \) \(\mathbf{v}(1)=5 \mathbf{j}\) \(\mathbf{r}(1)=\mathbf{0}\) Text Transcription: a(t)=t j+t k v(1)=5 j r(1)=0

In Exercises 23–28, use the given acceleration function to find the velocity and position vectors. Then find the position at time t=2. \(\mathbf{a}(t)=-32 \mathbf{k} \) \(\mathbf{v}(0)=3 \mathbf{i}-2 \mathbf{j}+\mathbf{k}\), \(\mathbf{r}(0)=5 \mathbf{j}+2 \mathbf{k}\) Text Transcription: a(t)=-32 k v(0)=3 i-2 j+k r(0)=5 j+2 k

In Exercises 23–28, use the given acceleration function to find the velocity and position vectors. Then find the position at time t=2. \(\mathbf{a}(t)=-\cos t \mathbf{i}-\sin t \mathbf{j} \) \(\mathbf{v}(0)=\mathbf{j}+\mathbf{k}\), \(\mathbf{r}(0)=\mathbf{i}\) Text Transcription: a(t)=-\cos t i-\sin t j v(0)=j+k r(0)=i

In Exercises 23–28, use the given acceleration function to find the velocity and position vectors. Then find the position at time t=2. \(\mathbf{a}(t)=e^{t} \mathbf{i}-8 \mathbf{k}\) \(\mathbf{v}(0)=2 \mathbf{i}+3 \mathbf{j}+\mathbf{k}\), \(\mathbf{r}(0)=\mathbf{0}\) Text Transcription: a(t)=e^t i-8 k v(0)=2 i+3 j+k r(0)=0

Projectile Motion In Exercises 29–44, use the model for projectile motion, assuming there is no air resistance. Find the vector-valued function for the path of a projectile launched at a height of 10 feet above the ground with an initial velocity of 88 feet per second and at an angle of \(30^{\circ}\) above the horizontal. Use a graphing utility to graph the path of the projectile Text Transcription: 30^circ

Projectile Motion In Exercises 29–44, 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^circ

Projectile Motion In Exercises 29–44, 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^circ

Projectile Motion In Exercises 29–44, 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^circ

Projectile Motion In Exercises 29–44, use the model for projectile motion, assuming there is no air resistance. Eliminate the parameter t from the position function 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 29–44, 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\(x^{2}\) Use the result of Exercise 33 to find the position function. Then find the speed and direction of the ball at the point at which it has traveled 60 feet horizontally. Text Transcription: x^2

Projectile Motion In Exercises 29–44, use the model for projectile motion, assuming there is no air resistance. Modeling Data After the path of a ball thrown by a baseball player is videotaped, it is analyzed on a television set with a grid covering the screen. The tape is paused three times and the positions of the ball are measured. The coordinates are approximately (0, 6.0), (15, 10.6), and (30, 13.4). (The x-coordinate measures the horizontal distance from the player in feet and the y-coordinate measures the height in feet.) (a) Use a graphing utility to find a quadratic model for the data. (b) Use a graphing utility to plot the data and graph the model. (c) Determine the maximum height of the ball. (d) Find the initial velocity of the ball and the angle at which it was thrown.

Projectile Motion In Exercises 29–44, 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. Use a graphing utility to graph the path of the ball and determine whether it will clear a 10-foot-high fence located 375 feet from home plate. Text Transcription: 22^circ

Projectile Motion In Exercises 29–44, 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 feet 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 \(\theta_{0}=10^{\circ}, \theta_{0}=15^{\circ}, \theta_{0}=20^{\circ}\) 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^circ theta_0=15^circ theta_0=20^circ theta_0=25^circ

Projectile Motion In Exercises 29–44, use the model for projectile motion, assuming there is no air resistance. 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^circ

Projectile Motion In Exercises 29–44, 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^circ

Projectile Motion In Exercises 29–44, 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 29–44, 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 29–44, 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^circ

Projectile Motion In Exercises 29–44, 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 ft/sec (b) \(\theta=10^{\circ}, v_{0}\)= 146 ft/sec (c) \(\theta =45^{\circ}, v_{0}\)=66 ft/sec (d) \(\theta=45^{\circ}, v_{0}\)= 146 ft/sec (e) \(\theta =60^{\circ}, v_{0}\)=66 ft/sec (f) \(\theta=60^{\circ}, v_{0}\)= 146 ft/sec Text Transcription: theta =10^circ v_0 theta =45^circ theta =60^circ

Projectile Motion In Exercises 29–44, 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 45 and 46, use the model for projectile motion, assuming there is no air resistance. [a(t) = -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^circ

Projectile Motion In Exercises 45 and 46, use the model for projectile motion, assuming there is no air resistance. [a(t) = -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^circ

Cycloidal Motion In Exercises 47 and 48, consider the motion of a point (or particle) on the circumference of a rolling circle. As the circle rolls, it generates the cycloid \(\mathbf{r}(t)=b(\omega t-\sin \omega t) \mathbf{i}+b(1-\cos \omega t) \mathbf{j}\) 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

Cycloidal Motion In Exercises 47 and 48, consider the motion of a point (or particle) on the circumference of a rolling circle. As the circle rolls, it generates the cycloid \(\mathbf{r}(t)=b(\omega t-\sin \omega t) \mathbf{i}+b(1-\cos \omega t) \mathbf{j}\) 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 1 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

Circular Motion In Exercises 49–52, 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 \(\omega\) = du/dt 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 ti+b sin omega tj omega

Circular Motion In Exercises 49–52, 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 \(\omega\) = du/dt 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 ti+b sin omega tj omega

Circular Motion In Exercises 49–52, 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 \(\omega\) = du/dt 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 ti+b sin omega tj omega

Circular Motion In Exercises 49–52, 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 \(\omega\) = du/dt is the constant angular velocity. Show that the magnitude of the acceleration vector is b\(\omega^{2}\) Text Transcription: r(t)=b cos omega ti+b sin omega tj omega omega^2

Circular Motion In Exercises 53 and 54, use the results of Exercises 49–52. 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: 1/32

Circular Motion In Exercises 53 and 54, use the results of Exercises 49–52 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.

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 \(\frac{v_{0}^{2} \cos \theta}{g}\left(\sin \theta+\sqrt{\sin ^{2} \theta+\frac{2 g h}{v_{0}^{2}}}\right)\) feet. Text Transcription: v_0 Theta t=v_0 sin theta+sqrtv_0^2 sin ^2 theta+2 g h/g r(t)=(v_0 cos theta) t i+[h+(v_0 sin theta) t-1/2 g t^2] j v_0^2 cos theta/g(sin theta+sqrtsin ^2 theta+2 g h/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. Find the total time of travel and the total horizontal distance traveled. Text Transcription: v_0 theta=42.5^circ

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

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.

Consider a particle moving on an elliptical path described by r(t) = a cos \(\omega\)ti+ b sin \(\omega\)tj, where \(\omega\) = d\(\theta\)/dt 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: omega theta

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

What is known about the speed of an object if the angle between the velocity and acceleration vectors is (a) acute and (b) obtuse?

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

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 t j. Show that the path of the object is a circle.

True or False? In Exercises 65–68, 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 65–68, 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 65–68, 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 65–68, 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.