Question

Frictionless Cart A small frictionless cart, attached to the wall by a spring, is pulled 10 cm from its rest position and released at time \(t=0\) to roll back and forth for 4 sec. Its position at time \(t\) is \(s=10 \cos \pi t .\) (a) What is the cart's maximum speed? When is the cart moving that fast? Where is it then? What is the magnitude of the acceleration then? (b) Where is the cart when the magnitude of the acceleration is greatest? What is the cart's speed then?

Short answer

The cart's maximum speed is \(10 \pi\) cm/s and it reaches this speed at \(t = 1/2\) and \(t = 3/2\) sec. At those times it's at equilibrium position and the acceleration is \(10 \pi^2\) cm/s². The acceleration is greatest, \(10 \pi^2\) cm/s², when the cart is at its extreme positions, 10 cm from equilibrium, where its speed is zero.

Step-by-step solution

Derive the Function of Velocity

To find the maximum speed, think of speed as the magnitude of the velocity. We can derive velocity as the derivative of the position function. Let's differentiate the position function \(s=10 \cos \pi t\), we get \(v = -10 \pi \sin \pi t\).

Find the Maximum Speed and the Time When it's Reached

The magnitude of the velocity gives us the speed. For any \(t\), the maximum value of \(\sin \pi t\) (and hence of \(\sin \pi t\)) is 1, so the maximum speed is \(10 \pi\) cm/s. The \(\sin \pi t\) obtains its maximum value when its argument \(\pi t\) equals \( \frac{\pi}{2}\), or \( \frac{3\pi}{2}\), 't' can be either \(\frac{1}{2}\) or \(\frac{3}{2}\) sec for a 4-sec interval.

Find the Position of Cart and the Magnitude of Acceleration at Maximum Speed

To find where the cart is at maximum speed, we substitute \(t = \frac{1}{2}\) and \(t = \frac{3}{2}\) into our position function. We find both times give \(s = 0\). To find the acceleration when cart is at maximum speed, we need to derive the velocity function, that gives us \(a = -(10 \pi^2) \cos \pi t\). Substituting \(t = \frac{1}{2}\) and \(t = \frac{3}{2}\) into our acceleration function, we find both times give \(a = 10 \pi^2\) cm/s².

Find the Position and Speed When Acceleration is Maximum

The magnitude of the acceleration reaches maximum value when value of \(\cos \pi t\) is maximum, i.e. 1. This happens when \(t = 0\), or \(2\), so at both ends of the 4-sec interval. In both cases, the cart is at its extreme position, 10 cm from the equilibrium position. The cart's speed at those times is zero because it's momentarily at rest.