A ball bearing is placed on an inclined plane

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Question from Leah, a student:

A ball bearing is placed on an inclined plane and begins to roll.
The angle of elevation of the plane is x.
The distance (in meters) that the ball bearing rolls in t seconds is s(t) = 4.9(sin x)t^2.
What is the speed of the ball bearing,
and what value of x will produce the maximum speed at a particular time?

Hi Leah,

For the first question what is the speed of the ball bearing think of x as fixed. That is you set up the inclined plane with some angle x, maybe 30 degrees, and roll the ball bearing down the plane. The displacement function is

s(t) = 4.9 (sin x) t² metres

The velocity function is the derivative of the displacement function with respect to time so to answer the first question find

v(t) = s'(t) metres per second.

For the second question you are asked to change the roles of x and t. You are at a particular time, maybe t = 5 seconds, and you are asked to find the value of x that maximizes the speed. This is a max-min problem. Write the expression for the velocity again but this time call it f(x) to emphasize that x is the variable and t is a constant. Differentiate again, this tie with respect to x to find f '(x) and use the derivative to find the value of x that maximizes f(x).

If you need further assistance write back,
Penny

Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.