A stream of water strikes a stationary turbine bladehorizontally, as the drawing illustrates. The incident water streamhas a velocity of +18.0 m/s, wh

Find the tension in each cord in Figure if the weight ofthe suspended object is w.
a)
b)

A rocket is fired in deep space, where gravity is negligible. In the first second it ejects of ${\displaystyle \frac{1}{160}}$ its mass as exhaust gas and has an acceleration of ${\displaystyle 16.0\frac{m}{s^2}}$.
What is the speed of the exhaust gas relative to the rocket?

An exceptional standing jump would raise a person 0.80 m off the ground. To do this, what force must a 66-kg person exert against the ground? Assume the person crouches a distance of 0.20 m prior to jumping, and thus the upward force has this distance to act over before he leaves the ground.

A tow truck pulls a car that is stuck in the mud, with a force of 2500 N as in Fig P5.27 The tow cable is under tension and therefor pulls downward and to the left on the pin at its upper end. The light pin is held in equilibrium by forces exerted by the two bars A and B. Each bar is a strut that is, each is a bar whose weight is small compared to the forces it exerts, and which exerts forces only through hinge pins at its ends. Each strut exerts a force directed parallel to its length. Determine the force of tension or compression in each strut. Proceed as follows Make a guess as to which way (pushing or pulling ) each force acts on the top pin Draw a free-body diagram of the pin. Use the condition for equilibrium of the pin to translate the free-body diagram into equations. From the equations calculate the forces exerted by struts A and B. If you obtain a positive answer, you correctly guessed the direction of the force. A negative answer means the direction should be reversed, but the absolute value correction should be reversed, but the absolute value correctly gives the magnitude of the force. If a strut pulls of a pin, it is in tension. If it pushes, the strut is in compression. Identify whether each strut is in tension or in compression.

Prove the following limit.
$lim5x-4=6$
$x\to 2$
SOLUTION 1. Preliminary analysis of the problem guessing a value for $\delta$ Let € be a given positive number.
We want to find a number $\delta$ such that
if $0\mid <x-2\mid <\delta \text{then}\mid (5x-4)-6\mid <ϵ$.
But $\mid (5x-4)-6\mid =\mid 5x-10\mid =5||$. Therefore, we want $\delta$
if $0<\mid x-2\mid <\delta \text{then}5\mid \mid <ϵ$
such that
that is, if $0<\mid x-2\mid <\delta \text{then}\mid \mid <ϵ\frac{ϵ}{5}$
This suggests that we should choose $\delta =ϵ/5$.
2. Proof (showing that $\delta$ works). Given $ϵ>0$, choose $\delta =$ 
$ϵ/5.\text{If}0<\mid \mid <\delta$, then
$\mid (5x-4)-6\mid =\mid \mid$
$=5\mid \mid <5\delta$
$=5\left(\right)$
$=ϵ$
Thus
if $0<\mid x-2\mid <\delta \text{then}\mid (5x-4)-6\mid <ϵ.$
Therefore, by the definition of a limit
$lim5x-4=6$
$x\to 2$

Prove that there is a positive integer that equals the sum of the positive integers not exceeding it. Is your proof constructive or nonconstructive?