Question

A ball bearing is placed on an inclined plane and begins to roll. The angle of elevation of the plane is \(\theta .\) The distance (in meters) the ball bearing rolls in \(t\) seconds is \(s(t)=4.9(\sin \theta) t^{2}\) (a) Determine the speed of the ball bearing after \(t\) seconds. (b) Complete the table and use it to determine the value of \(\theta\) that produces the maximum speed at a particular time. $$ \begin{array}{|l|l|l|l|l|l|l|l|} \hline \boldsymbol{\theta} & 0 & \pi / 4 & \pi / 3 & \pi / 2 & 2 \pi / 3 & 3 \pi / 4 & \pi \\ \hline \boldsymbol{s}^{\prime}(\boldsymbol{t}) & & & & & & & \\ \hline \end{array} $$

Short answer

The speed of the ball bearing after \(t\) seconds is \(9.8(\sin \theta)t\) m/s. The angle of elevation \(\theta\) that produces the maximum speed at a particular time is \(\pi/2\) radians or 90 degrees.

Step-by-step solution

Find the derivative

The derivative of \(s(t)\) with respect to \(t\) is \(s'(t)\). Since \(s(t)=4.9(\sin \theta) t^{2}\), the first step is to find its derivative using the power rule for derivatives which states \(d/dx [x^n] = nx^{n-1}\). Thus the derivative of \(s(t)\) is \(s'(t) = 4.9(\sin \theta) * 2t = 9.8(\sin \theta)t\).

Filling in the table

The next step is to substitute \(\theta\) values into \(s'(t)\) to obtain the values for the table. Complete the table by substituting 0, \( \pi / 4\), \( \pi / 3\), \( \pi / 2\), \(2 \pi / 3\), \(3 \pi / 4\), and \(\pi\) for \(\theta\) in the derivative function. For all \(t>0\), the maximum value of \(\sin \theta\) is 1, which is achieved at \(\theta = \pi / 2\). Therefore, \(\pi / 2\) produces the maximum speed.

Find the sin values

To complete the table, find the sine for each given angle. For example, \( \sin(0) = 0\), \( \sin(\pi / 4) = 0.707\), \( \sin(\pi / 3) = 0.866\), \( \sin(\pi / 2) = 1\), \( \sin(2 \pi / 3) = 0.866\), \( \sin(3 \pi / 4) = 0.707\), \( \sin(\pi) = 0\). Therefore, when the angle \(\theta = \pi / 2\), it maximizes the value of \( \sin(\theta) = 1\), hence resulting in the maximum speed for the ball bearing.

Find the maximum speed

The maximum speed is given by \(s'(\pi/2)*t = 9.8(\sin \pi/2)t = 9.8(1)t = 9.8t\). This indicates that the speed grows linearly with time and the maximum value is reached when \(\theta = \pi / 2\).