You toss a ball straight up, and it reaches a maximum height h above the launch position. If you want to double the length of the time interval during which the ball stays in the air above the launch position, to what maximum height do you have to throw it?

Rectangle ABCD with vertices A(3,-1), B(3,-2), C(1,-2), and D(1,-1) is reflected across the y-axis, and then its image is reflected across the x-axis. Describe a single transformation that moves the rectangle from its starting position to its final position.

The $2400-\mathrm{lb}$ car is traveling $40 \mathrm{mi} / \mathrm{h}$ at position 1 . If the combined effect of the aerodynamic drag on the car and the tangential force exerted on its wheels by the road is that they exert no net tangential force on the car, what is the magnitude of the car's velocity at position 2 ?

Find the velocity, speed, and acceleration of a particle moving with position function

$$\mathbf{r}(t)=\left(2 t^{2}-3\right) \mathbf{i}+2 t \mathbf{j}.$$

Sketch the path of the particle and draw the position, velocity, and acceleration vectors for t = 1.

For the velocity-versus-time graph of as we discussed earlier:
a. Draw the corresponding position-versus-time graph. Assume that $x=0 \mathrm{~m}$ at $t=0 \mathrm{~s}$.
b. What is the object's position at $t=12 \mathrm{~s}$ ?
c. Describe a moving object that could have these graphs.

Find the position vector $\mathbf{R}(t),$ given the velocity $\mathbf{V}(t),$ and the initial position $\mathbf{R}(0)$.

$$\mathbf{V}(t)=2 \sqrt{t} \mathbf{i}+(\cos t) \mathbf{j} ; \mathbf{R}(0)=\mathbf{i}+\mathbf{j}$$

In the following exercise, use the given acceleration function to find the velocity and position vectors. Then find the position at time $t=2$.

$$\mathbf{a}(t)=2 \mathbf{i}+3 \mathbf{k}$$

$$\mathbf{v}(0)=4 \mathbf{j}, \quad \mathbf{r}(0)=\mathbf{0}$$

Consider the following position functions position functions. a. Find the velocity and speed of the object. b. Find the acceleration of the object. $\mathbf{r}(t)=\left\langle t^{2}+3, t^{2}+10, \frac{1}{2} t^{2}\right\rangle$, for $t \geq 0$

Show that the period of the simple pendulum is

$$T =2 \pi \sqrt { L / g }.$$

Determine the length of a pendulum that takes one second to swing from its extreme position on the right to its extreme position on the left. Let

$$g = 9.8 \text { meters/second } ^ { 2 }.$$

A particle moves along the $x$-axis, its position at time t given by $x(t)=t /\left(1+t^2\right), t \geqslant 0$, where $t$ is measured in seconds and x in meters.

Graph the position, velocity, and acceleration functions for $0 \leqslant t \leqslant 4$.

Solve each problem. See the example in this section.

Refer to the previous exercise. Suppose that the initial position of the frog is $(0,4)$ and its landing position is $(6,0)$. The height of the frog is given by

$$h(x)=-\dfrac{1}{3} x^2+\dfrac{4}{3} x+4.$$

What was the maximum height?

A particle moves along the x-axis so that its acceleration function is $a(t)=-4 t+\sin \pi t .$ The velocity at $t=\frac{1}{2}$ is 4, and the position when t = 1 is 4. (a) Find the velocity and position functions for the particle. (b) Find v(0) and x(0).

In Fig. 22–12, assume that $C$ is completely discharged while in Position $2$. What is the voltage across $R_1$ exactly two time constants after $S_1$ is moved to Position $1$? a. $37\text{ V}$. b. $13.5\text{ V}$. c. $50\text{ V}$. d. $86\text{ V}$.

A position function is given, where t is in seconds and s is in meters. (a) Find the position at the given t-value. (b) Find the average velocity over the given interval. (c) Find the velocity at the given t-value. $s=2 t^{2}-4 t ;[7,7.5] ; t=7$