Question

1. Knight 2.20 Ball bearings are made by letting spherical drops of molten metal fall inside a tall tower – called a shot tower – and solidify as they fall.

   1. If a bearing needs 4.0 s to solidify enough for impact, how high must the tower be?

   2. What is the bearing’s impact velocity?

2, . For Grading Knight 2.57 A lead ball is dropped into a lake from a diving board 5.0 m above the water surface. After entering the water, it sinks to the bottom with a constant velocity equal to the velocity with which it hit the water. The ball reaches the bottom in 3.0 s after it is released. How deep is the lake?

Answer 1

1. Kinematic Equations:

The kinematic equations are the mathematical expressions that relate five kinematic parameters (initial velocity, final velocity, acceleration, time, displacement). These equations are used only in the case of uniformly accelerated motion.

2. Vertical Motion Under the Influence of Gravity:

An object dropped or thrown vertically from a height near the surface of the earth will be pulled down by gravity. If we ignore the air resistance or any other forces present, then the object has a constant vertical acceleration (the acceleration due to gravity g). Using the well known kinematic equations for constant acceleration we can analyze the motion of the object.

The height above water from which the lead ball is dropped is:  h = 5 m

We must first determine at which velocity the ball hits the water. To do so we will:

a) Assume no air resistance.

b Use the Law of conservation of mechanical energy: E=K+P

Where

E is the mechanical energy (which is constant)

K is the kinetic energy.

P is the potential energy.

With this we have:

Where:

m is the balls's mass <- we will see that it cancels out and as such we don't need to know it.

v is the speed when it hits the water.

g is the gravitational constant (we will assume g=9.8 m/s2 )

h is the height from which the ball fell.

Because when we initially drop the ball, all its energy is potential (and P= -mgh) and when it hits the water, all its energy is kinetic (K=m/2(v2)) and all that potential was converted to kinetic energy.

So,

The time taken by the ball to reach the bottom of the lake after it is dropped is: t = 3.0 s

The time taken by the lead ball to hit the lake surface is:

Then the time taken by the ball to sink to the bottom of the lake from the surface is:

t2 = 3-1.010 = 1.99 s

Therefore the depth of the lake is: