A toy gun uses a spring with a force constant of 300 N/m to propel a 10.0-g steel ball. If the spring is compressed 7.00 cm and friction is negligible: (a) How much force is needed to compress the spring? (b) To what maximum height can the ball be shot? (c) At what angles above the horizontal may a child aim to hit a target 3.00 m away at the same height as the gun? (d) What is the gun's maximum range on level ground?

(a) Calculate the energy in kJ used by a 55.0-kg woman who does 50 deep knee bends in which her center of mass is lowered and raised 0.400 m. (She does work in both directions.) You may assume her efficiency is 20%. (b) What is the average power consumption rate in watts if she does this in 3.00 min?

Calculate the power output needed for a 950-kg car to climb a 2.00° slope at a constant 30.0 m/s while encountering wind resistance and friction totaling 600 N.

Using values from the Table, how many DNA molecules could be broken by the energy carried by a single electron in the beam of an old-fashioned TV tube? (These electrons were not dangerous in themselves, but they did create dangerous x rays. Later model tube TVs had shielding that absorbed x rays before they escaped and exposed viewers.)

$$\begin{array}{|l|l|} \hline {\text { Object/phenomenon }} & {\text { Energy in joules }} \\ \hline \text { Big Bang } & 10^{68} \\ \hline \text { Energy released in a supernova } & 10^{44} \\ \hline \text { Fusion of all the hydrogen in Earth's oceans } & 10^{34} \\ \hline \text { Annual world energy use } & 4 \times 10^{20} \\ \hline \text { Large fusion bomb (9 megaton) } & 3.8 \times 10^{16} \\ \hline \text { 1 kg hydrogen (fusion to helium) } & 6.4 \times 10^{14} \\ \hline \text { 1 kg uranium (nuclear fission) } & 8.0 \times 10^{13} \\ \hline \text { Hiroshima-size fission bomb (10 kiloton) } & 4.2 \times 10^{13} \\ \hline 90,000 \text { -ton aircraft carrier at 30 knots } & 1.1 \times 10^{10} \\ \hline \text { 1 barrel crude oil } & 5.9 \times 10^{9} \\ \hline \text { 1 ton TNT } & 4.2 \times 10^{9} \\ \hline \text { 1 gallon of gasoline } & 1.2 \times 10^{8} \\ \hline \text { Daily home electricity use (developed countries) } & 7 \times 10^{7} \\ \hline \text { Daily adult food intake (recommended) } & 1.2 \times 10^{7} \\ \hline 1000-\mathrm{kg} \text { car at } 90 \mathrm{km} / \mathrm{h} & 3.1 \times 10^{5} \\ \hline 1 \mathrm{g} \text { fat }(9.3 \mathrm{kcal}) & 3.9 \times 10^{4} \\ \hline \text { ATP hydrolysis reaction } & 3.2 \times 10^{4} \\ \hline 1 \mathrm{g} \text { carbohydrate }(4.1 \mathrm{kcal}) & 1.7 \times 10^{4} \\ \hline 1 \mathrm{g} \text { protein }(4.1 \mathrm{kcal}) & 1.7 \times 10^{4} \\ \hline \text { Tennis ball at } 100 \mathrm{km} / \mathrm{h} & 22 \\ \hline \text { Mosquito }\left(10^{-2} \mathrm{g} \operatorname{at} 0.5 \mathrm{m} / \mathrm{s}\right) & 1.3 \times 10^{-6} \\ \hline \text { single electron in a } \mathrm{TV} \text { tube beam } & 4.0 \times 10^{-15} \\ \hline \text { Energy to break one DNA strand } & 10^{-19} \\ \hline \end{array}$$

A 60kg skier with an initial speed of 12 m/s coasts up a 2.5 m high rise that makes an angle of 35° with the horizontal. a) if the coefficient of friction between the skis and the snow is 0.08, find the speed when he reaches the top of the rise. b) how much work does friction do on the skier?

(a) Calculate the force needed to bring a 950-kg car to rest from a speed of 90.0 km/h in a distance of 120 m (a fairly typical distance for a non-panic stop). (b) Suppose instead the car hits a concrete abutment at full speed and is brought to a stop in 2.00 m. Calculate the force exerted on the car and compare it with the force found in part (a).

(a) Calculate the work done on a 1500-kg elevator car by its cable to lift it 40.0 m at constant speed, assuming friction averages 100 N. (b) What is the work done on the lift by the gravitational force in this process? (c) What is the total work done on the lift?

A swimmer exerts an average horizontal backward force of 80.0 N with his arm during each 1.80 m long stroke. (a) What is his work output in each stroke? (b) Calculate the power output of his arms if he does 120 strokes per minute.

The recursive definition, a(n+1)=ra(n), $a(0)=a_{0}$, is called a first-order difference equation and generates the sequence $a_{0}, r a_{0}, r\left(r a_{0}\right), r\left(r\left(r a_{0}\right)\right)$, .... A little simplification shows that the nth term of this sequence is $a(n)=a_{0} r^{n}$. Now suppose that I represents the annual interest rate, but the interest is awarded in discrete packets, m times per year. Then the rate awarded during each compounding period is I/m. Consequently, if the initial investment is $P_{0}$, the balance is $P_{0}(1+I / m)$ at the end of the first compounding period, $P_{0}(1+I / m)^{2}$ at the end of the second compounding period, and so on. (a) Give a first-order difference equation with an initial condition that generates a sequence describing the balance in the account at the end of each compounding period. (b) Find a formula for the nth term of the sequence generated by the first-order difference equation created in part (a).

Find the indefinite integral. $\int 2\left(x^{2}-1\right)^{9} x d x$

A physical pendulum is made of a uniform disk of mass M and radius R suspended from a rod of negligible mass. The distance from the pivot to the center of the disk is $l.$ What value of $l$ makes the period a minimum?

Let f(x) be a monic polynomial of degree n with roots

$$\alpha_{1}, \alpha_{2}, \ldots,\alpha_{n}.$$

(a) Show that the discriminant D of f(x) is the square of the Vandermonde determinant

$$\left|\begin{array}{ccccc} 1 & \alpha_{1} & \alpha_{1}^{2} & \dots & \alpha_{1}^{n-1} \\ 1 & \alpha_{2} & \alpha_{2}^{2} & \dots & \alpha_{2}^{n-1} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & \alpha_{n} & \alpha_{n}^{2} & \dots & \alpha_{n}^{n-1} \end{array}\right|=\prod_{i>j}\left(\alpha_{i}-\alpha_{j}\right)$$

(b) Taking the Vandermonde matrix above, multiplying on the left by its transpose and taking the determinant show that one obtains

$$\boldsymbol{D}=\left|\begin{array}{ccccc} p_{0} & p_{1} & p_{2} & \dots & p_{n-1} \\ p_{1} & p_{2} & p_{3} & \dots & p_{n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ p_{n-1} & p_{n} & p_{n+1} & \dots & p_{2 n-2} \end{array}\right|$$

where

$$p_{i}=\alpha_{1}^{i}+\cdots+\alpha_{n}^{i}$$

is the sum of the $i^{\text {th }}$ powers of the roots of f(x), which can be computed in terms of the coefficients of f(x) using Newton's formulas above. This gives an efficient procedure for calculating the discriminant of a polynomial.

For the following exercise, use the functions y = f(x) to find a. $\frac{d f}{d x}$ at x = a and b. $x=f^{-1}(y)$. c. Then use part b. to find $\frac{d f^{-1}}{d y}$ at y = f(a). $f(x)=2 x^{3}-3, x=1$

Let

$$\mathbf{u}=\left[\begin{array}{l} 0 \\ 1 \\ 3 \end{array}\right] \quad \mathbf{v}=\left[\begin{array}{r} 1 \\ -1 \\ 2 \end{array}\right]$$

$$\mathbf{w}=\left[\begin{array}{r} 1 \\ 1 \\ -3 \end{array}\right]$$

Compute the quantity. $\frac{\mathbf{u} \cdot \mathbf{w}}{\mathbf{w} \cdot \mathbf{w}} \mathbf{w}$