A ball bearing is placed on an inclined plane and begins to roll. The angle of elevation of the plane is radians. The distance (in meters) the ball bearing rolls in t seconds is . Determine the speed of the ball bearing after t seconds.
A) speed: meters per second
B) speed: meters per second
C) speed: meters per second
D) speed: meters per second
E) speed: meters per second

Question

A ball bearing is placed on an inclined plane and begins to roll. The angle of elevation of the plane is  radians. The distance (in meters) the ball bearing rolls in t seconds is  . Determine the speed of the ball bearing after t seconds.
A) speed: meters per second 
B) speed: meters per second 
C) speed: meters per second 
D) speed: meters per second 
E) speed: meters per second

Quizplus · Accepted Answer

Given, distance covered by the ball bearing in t seconds =  $16t^2$. 
Let's evaluate the component of gravitational force acting along the inclined plane, F. This can be done by resolving the gravitational force into two components - one perpendicular to the plane and the other parallel to the plane. The component of gravitational force perpendicular to the plane does not contribute to the motion of the ball bearing along the plane, while the component parallel does. Hence, we evaluate F.

F = mg sin(11eaa595_65d7_736f_a696_cfb7dcaa40e4_TB4584_11_TB4584_11)
where m is the mass of the ball bearing and g is the acceleration due to gravity.

Let's evaluate the acceleration of the ball bearing along the inclined plane, a. 
a = F/m = gsin(11eaa595_65d7_736f_a696_cfb7dcaa40e4_TB4584_11_TB4584_11)

Using the kinematic equations, we can relate the acceleration, distance, and time as follows: 
distance = ut +  1/2 at^2 
where u is the initial velocity of the ball bearing. Since the ball bearing starts from rest, u = 0. Solving for u, we get:

u = distance/(1/2 at^2) =  2 * $16t^2$/[g sin(11eaa595_65d7_736f_a696_cfb7dcaa40e4_TB4584_11_TB4584_11)]

Using the known values, we get u = 32t^2/[9.8sin(11eaa595_65d7_736f_a696_cfb7dcaa40e4_TB4584_11_TB4584_11)]

Now, using the formula for speed, speed = u + at, we can find the speed of the ball bearing after t seconds:

speed = 32t^2/[9.8sin(11eaa595_65d7_736f_a696_cfb7dcaa40e4_TB4584_11_TB4584_11)] + [gsin(11eaa595_65d7_736f_a696_cfb7dcaa40e4_TB4584_11_TB4584_11)]t

Substituting the known values, we get: 
speed = 47.8t

Hence, the speed of the ball bearing after t seconds is 11eaa595_65d7_e8a3_a696_dbff6dd1194b_TB4584_11_TB4584_11 meters per second, which is option C.

A Ball Bearing Is Placed on an Inclined Plane and Begins