Chapter 18

Suppose that a point moves on the circle of radius \(R\) with constant angular velocity \(\omega\). Find the tangential and normal accelerations. Ans. \(a_{T}=0, a_{N}=\omega^{2} R\).

Find \(d y / d x\) and \(d^{2} y / d x^{2}\) for each of the following parametric representations of the functions relating \(x\) and \(y\) : (a) \(x=2-t, y=t^{2}\). Ans. \(d y / d x=-2 t, d^{2} y / d x^{2}=2\). (b) \(x=3 t^{2}, y=4 t-5\). (c) \(x=t e^{-t}, y=e^{t}\). Ans. \(d y / d x=e^{2 t} /(1-t), d^{2} y / d x^{2}=e^{3 t}(3-2 t)(1\) \(-t)^{3}\).

Find parametric equations of the straight line which goes through the origin and makes an angle of \(45^{\circ}\) with the \(x\)-axis. Ans. \(x=(\sqrt{2} / 2) t, y=(\sqrt{2} / 2) t\).

In the following exercises draw the curve by working with the parametric equations. Then find the direct relation between \(x\) and \(y\) (a) \(x=2 t+1, y=1-t . \quad\) Ans. \(y=(3-x) / 2\). (b) \(x=t^{2}, y=2 t\). (c) \(x=2 \sin t, y=2 \cos t . \quad\) Ans. \(x^{2}+y^{2}=4\). (d) \(x=t^{2}, y=t^{2}-5 t\). (e) \(x=e^{t}, y=e^{-t}\) Suggestion: Use the e-table. Ans. \(y=1 / x, x>0\). (f) \(x=\cos t, y=2 \sin t\). (g) \(x=\cos ^{2} \pi t, y=\sin ^{2} \pi t . \quad\) Ans. \(x+y=1, x>0, y>0 .\) (h) \(x=\sin t, y=\sin t\). (i) \(x=e^{t}, y=e^{t}\).

Find a parametric representation of the straight line through the origin which makes an acute angle \(\beta\) with the \(x\)-axis.

For the cycloid \(x=R(\omega t-\sin \omega t), y=R(1-\cos \omega t)\) calculate the following: (a) The tangential acceleration at an arbitrary point \(P\) of the curve. Ans. \(R \omega^{2} \cos (\omega t / 2)\). (b) The normal acceleration of \(P\). Ans. \(-R \omega^{2} \sin (\omega t / 2)\). (c) The magnitude of the resultant acceleration. Ans. \(R \omega^{2}\).

A particle moves on the hyperbola whose parametric equations are \(x=\) \(r \cosh \omega t, y=r \sinh \omega t\), where \(\omega\) and \(r\) are constants. Calculate the velocity and acceleration vectors. Ans. \(v=(\omega r \sinh \omega t) \mathrm{i}+(\omega r\) \(\cosh \omega t) j ; a=\omega^{2} r\).

Could we use the parametric equations \(x=3 t, y=3 t\) to represent the straight line through the origin and making an angle of \(45^{\circ}\) with the \(x-\) axis? If so, and if \(t\) represents time, what would these equations say about motion along this line as compared with the motion represented by the parametric equations of Exercise \(1 ?\)

A particle moves along a curve with constant speed. Show that the acceleration is perpendicular to the velocity.

Suppose that a particle moves along a cycloid in accordance with \(\theta=\) \(\omega t\), where \(\omega\) is constant: (a) When is the magnitude of the velocity 0 and where is the particle at those instants? Ans. \(t=0\) or \(2 \pi / \omega\). (b) When is the magnitude of the velocity maximum? Because the generating circle rotates at a constant angular velocity, can you account in physical terms for the fact that \(P\) has a maximum magnitude of velocity at the point that you have determined? $$ \text { Ans. } t=\pi / \omega \text {. } $$

Construct the graph of the parametric equations \(x=2 \sin ^{2} \theta\) and \(y=2\) \(\cos ^{2} \theta\) .

A ball is thrown out horizontally with a velocity \(V\) from the top of a building standing on a horizontal plane. At the same instant another ball is dropped from the top. Show that both reach the ground at the same time.

For what angle of fire does a shell attain maximum height? Ans. \(90^{\circ} .\)

The motion of an object released by an airplane is given by the equations \(x=V t, y=h-16 t^{2}\). These equations presume that the \(x\)-axis is horizontal and the \(y\)-axis is directed upward and measured from the ground. Find the following: (a) The direct equation of the path. Ans. \(y=h-16\left(x^{2} / V^{2}\right)\). (b) The range, that is, the horizontal distance traveled by the object. Ans. \((V / 4) \sqrt{h}\). (c) In view of the answer to part (b), which factor is more important in increasing the range-the height \(h\) from which the object is released or the initial velocity \(V\) ?

A point on the rim of a train wheel follows a cycloidal path given by the equations \(x=20 t-\sin 20 t, y=1-\cos 20 t\). Determine the magnitude and direction of the acceleration of the point. Ans. 400 ; toward the center of the wheel.

A wheel 4 feet in diameter rolls along a straight horizontal road at the rate of \(20 \mathrm{rpm}\). Find the magnitude of the velocity and the acceleration of a point on the rim when it is 1 foot below the center of the wheel.

A ball is hit by a bat at a point \(h\) feet above the ground (Fig. 18-15) and the ball is projected with an initial velocity \(V\) and at angle \(A\) to the horizontal. Find the range \(r\), that is, find the horizontal distance from the starting point to the point at which the ball strikes the ground. Ans. \(\left(V^{2} / 64\right)\left(\sin 2 A+\sqrt{\sin ^{2} 2 A+\left(256 h / V^{2}\right) \cos ^{2} A}\right.\).