What force does a trampoline have to apply to Jennifer, a 45.0-kg gymnast, to accelerate her straight up at 7.50 m/s²? The answer is independent of the velocity of the gymnast - she can be moving up or down or can be instantly stationary.

A 1100-kg car pulls a boat on a trailer. (a) What total force resists the motion of the car, boat, and trailer, if the car exerts a 1900-N force on the road and produces an acceleration of $0.550\ \mathrm{m} / \mathrm{s}^{2} ?$ The mass of the boat plus trailer is 700 kg. (b) What is the force in the hitch between the car and the trailer if 80% of the resisting forces are experienced by the boat and trailer?

When starting a foot race, a 70.0-kg sprinter exerts an average force of 650 N backward on the ground for 0.800 s. (a) What is his final speed? (b) How far does he travel?

A $61 \mathrm{~kg}$ gymnast falls vertically from a jump onto a trampoline. Her speed as she hits the trampoline is $5.2 \mathrm{~m} / \mathrm{s}$, and she comes to a stop in $0.20 \mathrm{~s}$. What is the average magnitude of the force exerted on the gymnast by the trampoline? (a) $6.1 \mathrm{~N}$ (b) $305 \mathrm{~N}$ (c) $610 \mathrm{~N}$ (d) $1600 \mathrm{~N}$

Factor each trinomial, if possible.

$$10 a ^ { 2 } + 11 a + 3$$

A gymnast is bouncing on a trampoline. After a high bounce the gymnast comes down and hits the elastic surface of the trampoline. In so doing the gymnast applies a force to the trampoline. (a) Describe the effect this force has on the elastic surface. (b) The surface applies a reaction force to the gymnast. Describe the effect that this reaction force has on the gymnast.

Find a weighted Euclidean inner product on

$$R^n$$

such that the vectors

$$v_1 = (1, 0, 0, ... , 0),v_2 = (0, √2, 0, . .. , 0),v_3 = (0, 0, √3, ...,0),....., v_n = (0, 0, 0, ... ,√n)$$

form an orthonormal set.

A research firm tests the miles-per-gallon characteristics of three brands of gasoline. Because of different gasoline performance characteristics in different brands of automobiles, five brands of automobiles are selected and treated as blocks in the experiment; that is, each brand of automobile is tested with each type of gasoline. The results of the experiment (in miles per gallon) follow.

$$\begin{matrix} \text{ } & \text{ } & \text{Gasoline} & \text{Brands} & \text{ }\\ \text{ } & \text{ } & \text{I} & \text{II} & \text{III}\\ \text{ } & \text{A} & \text{18} & \text{21} & \text{20}\\ \text{ } & \text{B} & \text{24} & \text{26} & \text{27}\\ \text{Automobiles} & \text{C} & \text{30} & \text{29} & \text{34}\\ \text{ } & \text{D} & \text{22} & \text{25} & \text{24}\\ \text{ } & \text{E} & \text{20} & \text{23} & \text{24}\\ \end{matrix}$$

                a. At $\alpha =$ .05, is there a significant difference in the mean miles-per-gallon characteristics

of the three brands of gasoline? b. Analyze the experimental data using the ANOVA procedure for completely randomized designs. Compare your findings with those obtained in part (a). What is the advantage of attempting to remove the block effect?

Commercial airplanes are sometimes pushed out of the passenger loading area by a tractor. (a) An 1800-kg tractor exerts a force of $1.75 \times 10^{4}\ \mathrm{N}$ backward on the pavement, and the system experiences forces resisting motion that total 2400 N. If the acceleration is $0.150 \mathrm{m} / \mathrm{s}^{2},$ what is the mass of the airplane? (b) Calculate the force exerted by the tractor on the airplane, assuming 2200 N of the friction is experienced by the airplane. (c) Draw two sketches showing the systems of interest used to solve each part, including the free-body diagrams for each.

A 200-lb traffic light supported by two cables hangs in equilibrium. As shown in figure, let the weight of the light be represented by $\boldsymbol{w}$ and the forces in the two cables by $\mathbf{F}_1$ and $\mathbf{F}_2$. From figure, we see that a condition of equilibrium is

$$w+\mathrm{F}_1+\mathrm{F}_2=\mathbf{0}$$

If

$$\begin{aligned} & \mathbf{w}=-200 \mathbf{j} \\ & \mathbf{F}_1=\left(\left\|\mathbf{F}_1\right\| \cos 20^{\circ}\right) \mathbf{i}+\left(\left\|\mathbf{F}_1\right\| \sin 20^{\circ}\right) \mathbf{j} \\ & \mathbf{F}_2=-\left(\left\|\mathbf{F}_2\right\| \cos 15^{\circ}\right) \mathbf{i}+\left(\left\|\mathbf{F}_2\right\| \sin 15^{\circ}\right) \mathbf{j}, \end{aligned}$$

use (7) to determine the magnitudes of $\mathbf{F}_1$ and $\mathbf{F}_2$.

A cleaner pushes a 4.50-kg laundry cart in such a way that the net external force on it is 60.0 N. Calculate the magnitude of its acceleration.

(a) What net external force is exerted on a 1100.0-kg artillery shell fired from a battleship if the shell is accelerated at

$$2.40 × 10^4 m/s^2?$$

(b) What is the magnitude of the force exerted on the ship by the artillery shell, and why?

A cleaner pushes a 4.50-kg laundry cart in such a way that the net external force on it is 60.0 N. Calculate the magnitude of his cart's acceleration.

The weight of an astronaut plus his space suit on the Moon is only 250 N. (a) How much does the suited astronaut weigh on Earth? (b) What is the mass on the Moon? On Earth?