College Physics by Openstax Chapter 2 Problem 46: Kinematic analysis of a swimmer’s vertical motion from a diving board

Problem:

A swimmer bounces straight up from a diving board and falls feet first into a pool. She starts with a velocity of 4.00 m/s, and her takeoff point is 1.80 m above the pool.

(a) How long are her feet in the air?

(b) What is her highest point above the board?

(c) What is her velocity when her feet hit the water?

Solution:

The known values are: $y_0=1.80\:\text{m}$ , $y=0\:\text{m}$ , $a=-9.80\:\text{m/s}^2$ , $v_0=4.00\:\text{m/s}$ .

Part A

Based from the knowns, the formula most applicable to solve for the time is $\Delta y=v_0t+\frac{1}{2}at^2$ . If we rearrange the formula by solving for $t$ , and substitute the given values, we have

\begin{align*} t & =\frac{-v_0\pm \sqrt{v_0^2-2a\Delta y}}{a} \\ t & =\frac{-4.00\:\text{m/s}\pm \sqrt{\left(4.00\:\text{m/s}\right)^2-2\left(-9.80\:\text{m/s}^2\right)\left(1.80\:\text{m}\right)}}{-9.80\:\text{m/s}^2} \\ t & =1.14\:\text{s} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right) \end{align*}

Part B

We have the formula

\begin{align*} \Delta y & =\frac{v^2-v_0^2}{2a} \\ \Delta y & =\frac{\left(0\:\text{m/s}\right)^2-\left(4.00\:\text{m/s}\right)^2}{2\left(-9.80\:\text{m/s}^2\right)} \\ \Delta y & =0.816\: \text{m} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right) \end{align*}

Part C

The formula to be used is

v^2=v_0^2+2a\Delta y

Substituting the given values

\begin{align*} v^2 & =v_0^2+2a\Delta y \\ v & =\pm \sqrt{v_0^2+2a\Delta y} \\ v & =\pm \sqrt{\left(4.00\:\text{m/s}\right)^2+2\left(-9.80\:\text{m/s}^2\right)\left(-1.80\:\text{m}\right)} \\ v & =\pm \sqrt{51.28\:\text{m}^2/\text{s}^2} \\ v &=\pm 7.16\:\text{m/s} \end{align*}

Since the diver must be moving in the negative direction,

v=-7.16\:\text{m/s} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)

College Physics Chapter 2 Problems

Problem 1	Problem 2	Problem 3	Problem 4	Problem 5
Problem 6	Problem 7	Problem 8	Problem 9	Problem 10
Problem 11	Problem 12	Problem 13	Problem 14	Problem 15
Problem 16	Problem 17	Problem 18	Problem 19	Problem 20
Problem 21	Problem 22	Problem 23	Problem 24	Problem 25
Problem 26	Problem 27	Problem 28	Problem 29	Problem 30
Problem 31	Problem 32	Problem 33	Problem 34	Problem 35
Problem 36	Problem 37	Problem 38	Problem 39	Problem 40
Problem 41	Problem 42	Problem 43	Problem 44	Problem 45
Problem 46	Problem 47	Problem 48	Problem 49	Problem 50
Problem 51	Problem 52	Problem 53	Problem 54	Problem 55
Problem 56	Problem 57	Problem 58	Problem 59	Problem 60
Problem 61	Problem 62	Problem 63	Problem 64	Problem 65
Problem 66