3.12 . A rookie quarterback throws a football with an initial upward velocity component of 12.0 m/s and a horizontal velocity component of 20.0 m/s. Ignore air resistance. (a) How much time is required for the football to reach the highest point of the trajectory? (b) How high is this point? (c) How much time (after it is thrown) is required for the football to return to its original level? How does this compare with the time calculated in part (a)? (d) How far has the football traveled horizontally during this time? (e) Draw x-t, y-t, and graphs for the motion.
SOLUTION
From the figure, -y direction is downward. Consider the following
$v_{0x}=20.0 m/s $
$v_{0y}=12.0 m/s $
$a_x=0$, and $a_y=-g=-9.8 m/s^2$
at the highest point of the trajectory, $v_y=0$
a) Solving for the time,
$v_y=v_{0y}+a_yt$
$0=12.0 m/s + a_yt$
$-12.0 m/s = a_yt$
$\frac{-12.0 m/s}{-g} = t$
$t=1.22s$
b) We may choose from the following equations, but the answer is the same:
$y=y_0+v_{0y}t+\frac{1}{2}a_yt^2$
$v^2_y=v^2_{0y}+2a_y (y-y_0)$
$y=y_0+(\frac{v_{0y}+v_y}{2})t
From $y=y_0+v_{0y}t+\frac{1}{2}a_yt^2$:
$y=\frac{1}{2}a_yt^2=-\frac{1}{2}gt^2$
$y=-\frac{1}{2}(9.8 / m/s^2)(1.22s)^2=7.3 / m$
From $v^2_y=v^2_{0y}+2a_y(y-y_0)$
$0= v^2_{0y}-2gy$
$y=\frac{ v^2_{0y}}{2g}=7.3 / m$
From $y=y_0+(\frac{v_{0y}+v_y}{2})t
$y= 0+(\frac{12.0 / m/s / + 0}{2})(1.22s)=7.3 / m$
A daring 510-N swimmer dives off a cliff with a running horizontal leap, as shown in
3.10 A daring 510-N swimmer dives off a cliff with a running horizontal leap, as shown in Fig. E3.10. What must her minimum speed be just as she leaves the top of the cliff so that she will
miss the ledge at the bottom, which is 1.75 m wide and 9.00 m below the top of the cliff?
SOLUTION
The person moves in projectile motion once it takes off from the cliff. I will place my frame of reference at the edge of the cliff where –y-direction is downward.
miss the ledge at the bottom, which is 1.75 m wide and 9.00 m below the top of the cliff?
SOLUTION
The person moves in projectile motion once it takes off from the cliff. I will place my frame of reference at the edge of the cliff where –y-direction is downward.
Division of Mixed Numbers
To divide mixed numbers, first write them as fractions. Then divide the fractions. For example,
$2 \ \frac{1}{3}$ ÷ $3 \ \frac{4}{5}$
$= \ \frac{7}{3}$ ÷ $\frac{19}{5}$
$= \ \frac{7}{3}$ x $\frac{5}{19}$
$= \ \frac{7 \ x \ 5}{3 \ x \ 19}$
$= \ \frac{35}{57}$
We may also use the cancellation method. In the example below, the whole number is written as fraction and the mixed number to fraction. It is shown below:
$3$ ÷ $1 \ \frac{2}{7}$
$= \ \frac{3}{1}$ ÷ $\frac{9}{7}$
$= \ \frac{3}{1}$ x $\frac{7}{9}$
$= \ \frac{1 \ x \ 7}{1 \ x \ 3}$
$= \ \frac{7}{3}$
$2 \ \frac{1}{3}$ ÷ $3 \ \frac{4}{5}$
$= \ \frac{7}{3}$ ÷ $\frac{19}{5}$
$= \ \frac{7}{3}$ x $\frac{5}{19}$
$= \ \frac{7 \ x \ 5}{3 \ x \ 19}$
$= \ \frac{35}{57}$
We may also use the cancellation method. In the example below, the whole number is written as fraction and the mixed number to fraction. It is shown below:
$3$ ÷ $1 \ \frac{2}{7}$
$= \ \frac{3}{1}$ ÷ $\frac{9}{7}$
$= \ \frac{3}{1}$ x $\frac{7}{9}$
$= \ \frac{1 \ x \ 7}{1 \ x \ 3}$
$= \ \frac{7}{3}$
Adding and Subtracting Fractions with Unlike Denominators
When adding and subtracting fractions with unlike denominators then you need to find the least common denominator, LCD.
add: $\frac{1}{3} \ + \ \frac{1}{5}$
add: $\frac{1}{3} \ + \ \frac{1}{5}$
Dividing Whole Numbers and Fractions
To divide by a fraction, multiply by its reciprocal.
Examples of reciprocal of a number:
1. reciprocal of $\frac{1}{2}$ is $\frac{2}{1}$ or 2
2. reciprocal of $\frac{5}{13}$ is $\frac{13}{5}$
Let us divide whole number by a fraction:
$1$ ÷ $ \frac{7}{8}$ = $1$ x $ \frac{8}{7}$ = $\frac{8}{7}$ or $1 \ \frac{1}{7}$
$6$ ÷ $ \frac{3}{5}$ = $6$ x $ \frac{5}{3}$ = $\frac{2 \ x \ 5}{1}$ = $10$
$18$ ÷ $ \frac{2}{7}$ = $18$ x $ \frac{7}{2}$ = $\frac{9 \ x \ 7}{1}$ = $63$
From the examples,we use the reverse of division which is multiplication and write the reciprocal of the divisor, then multiply.
Remember that any number divided by zero does not exist. Also 0 has no reciprocal.
Now let us divide fractions:
$ \frac{3}{4}$ ÷ $ \frac{15}{16}$ = $\frac{3}{4}$ x $ \frac{16}{15}$ = $\frac{1 \ x \ 4}{1 \ x \ 5}$ = $\frac{4}{5}$
The reciprocal of the divisor $ \frac{15}{16}$ was used. We apply cancellation method then multiply.
Examples of reciprocal of a number:
1. reciprocal of $\frac{1}{2}$ is $\frac{2}{1}$ or 2
2. reciprocal of $\frac{5}{13}$ is $\frac{13}{5}$
Let us divide whole number by a fraction:
$1$ ÷ $ \frac{7}{8}$ = $1$ x $ \frac{8}{7}$ = $\frac{8}{7}$ or $1 \ \frac{1}{7}$
$6$ ÷ $ \frac{3}{5}$ = $6$ x $ \frac{5}{3}$ = $\frac{2 \ x \ 5}{1}$ = $10$
$18$ ÷ $ \frac{2}{7}$ = $18$ x $ \frac{7}{2}$ = $\frac{9 \ x \ 7}{1}$ = $63$
From the examples,we use the reverse of division which is multiplication and write the reciprocal of the divisor, then multiply.
Remember that any number divided by zero does not exist. Also 0 has no reciprocal.
Now let us divide fractions:
$ \frac{3}{4}$ ÷ $ \frac{15}{16}$ = $\frac{3}{4}$ x $ \frac{16}{15}$ = $\frac{1 \ x \ 4}{1 \ x \ 5}$ = $\frac{4}{5}$
The reciprocal of the divisor $ \frac{15}{16}$ was used. We apply cancellation method then multiply.
Least Common Multiple
Let us consider the list of multiples of 2, 3, and 4.
Multiples of 2: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, ...
Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, ...
Multiples of 4: 4, 8, 12, 16, 20, 24, ...
Multiples of 2: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, ...
Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, ...
Multiples of 4: 4, 8, 12, 16, 20, 24, ...
Addition and Subtraction of Fractions with Like Denominators
The illustration below show how to add and subtract fractions with like denominators:
To add or subtract fractions with like denominators, write the sum or difference of the numerators over the denominator.
$\frac{a}{c} + \frac{b}{c}=\frac{a+b}{c}$
$\frac{a}{c} - \frac{b}{c}=\frac{a-b}{c}$
Examples:
1. $\frac{4}{6} + \frac{2}{6}=\frac{6}{6}=1$
2. $\frac{6}{11} - \frac{3}{11}=\frac{3}{11}$
3. $\frac{7}{3} - \frac{2}{3}=\frac{5}{3} $ or $1\frac{2}{3}$
4. $\frac{1}{8} + \frac{4}{8} + \frac{2}{8}=\frac{7}{8}$
5. $(\frac{10}{13} + \frac{2}{13}) - \frac{7}{13}=\frac{5}{13}$
Always write your answer in lowest term. If for instance your numerator is greater than the denominator then write the answer as a mixed number like in example 3 or if they are equal then you get a whole number like example 1. When you see a parenthesis, work inside it first like in example 5.
To add or subtract fractions with like denominators, write the sum or difference of the numerators over the denominator.
$\frac{a}{c} + \frac{b}{c}=\frac{a+b}{c}$
$\frac{a}{c} - \frac{b}{c}=\frac{a-b}{c}$
Examples:
1. $\frac{4}{6} + \frac{2}{6}=\frac{6}{6}=1$
2. $\frac{6}{11} - \frac{3}{11}=\frac{3}{11}$
3. $\frac{7}{3} - \frac{2}{3}=\frac{5}{3} $ or $1\frac{2}{3}$
4. $\frac{1}{8} + \frac{4}{8} + \frac{2}{8}=\frac{7}{8}$
5. $(\frac{10}{13} + \frac{2}{13}) - \frac{7}{13}=\frac{5}{13}$
Always write your answer in lowest term. If for instance your numerator is greater than the denominator then write the answer as a mixed number like in example 3 or if they are equal then you get a whole number like example 1. When you see a parenthesis, work inside it first like in example 5.
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