Question

problem 1. a squirrel climbs to the top of a 28 - m tall tree with an initial velocity of 6 m/s. if the final velocity of the squirrel is 4.3 m/s, calculate the amount of time it took the squirrel to climb to the top of the tree.
problem 2. a swan is swimming in a pond with an initial velocity of 1 m/s. it travels a displacement of 20 m and reaches a final velocity of 3 m/s. determine how much time it took the swan to travel this far.
problem 3. a typhoon is accelerating at a rate of 1.3 m/s² for 12 s. if the final velocity of the typhoon is 30 m/s, determine the initial velocity.
problem 4. a horse is walking with an initial velocity of 5 m/s. after slowing down for 25 s, the horse comes to a complete stop. calculate the acceleration of the horse.
problem 5. a baseball is hit with an initial velocity of 14.4 m/s. it travels a displacement of 132 m in 11 s. calculate the final velocity of the baseball.

Explanation:

Problem 1:

Step1: Identify the kinematic - equation

We use the equation $v = v_0+at$ and $d=v_0t+\frac{1}{2}at^{2}$, but we can also use the equation $v^{2}-v_0^{2}=2ad$. First, find the acceleration using $v^{2}-v_0^{2}=2ad$. Given $v_0 = 6\ m/s$, $v = 4.3\ m/s$ and $d = 26\ m$. Rearranging for $a$ gives $a=\frac{v^{2}-v_0^{2}}{2d}$. Then use $v = v_0+at$ to find $t$.
$a=\frac{4.3^{2}-6^{2}}{2\times26}=\frac{18.49 - 36}{52}=\frac{- 17.51}{52}\approx - 0.337\ m/s^{2}$

Step2: Calculate the time

Using $v = v_0+at$, we can solve for $t$. Rearranging gives $t=\frac{v - v_0}{a}$. Substituting $v = 4.3\ m/s$, $v_0 = 6\ m/s$ and $a=-0.337\ m/s^{2}$, we get $t=\frac{4.3 - 6}{-0.337}=\frac{-1.7}{-0.337}\approx5.04\ s$

Problem 2:

Step1: Use the kinematic - equation

We use the equation $v^{2}-v_0^{2}=2ad$ to find the acceleration $a$. Given $v_0 = 1\ m/s$, $v = 3\ m/s$ and $d = 20\ m$. Rearranging for $a$ gives $a=\frac{v^{2}-v_0^{2}}{2d}=\frac{3^{2}-1^{2}}{2\times20}=\frac{9 - 1}{40}=\frac{8}{40}=0.2\ m/s^{2}$

Step2: Calculate the time

Using $v = v_0+at$, we can solve for $t$. Rearranging gives $t=\frac{v - v_0}{a}$. Substituting $v = 3\ m/s$, $v_0 = 1\ m/s$ and $a = 0.2\ m/s^{2}$, we get $t=\frac{3 - 1}{0.2}=\frac{2}{0.2}=10\ s$

Problem 3:

Step1: Use the kinematic - equation

We use the equation $v = v_0+at$. Given $a = 1.3\ m/s^{2}$, $t = 12\ s$ and $v = 32\ m/s$. Rearranging for $v_0$ gives $v_0=v - at$.
$v_0=32-1.3\times12=32 - 15.6 = 16.4\ m/s$

Problem 4:

Step1: Use the kinematic - equation

We use the equation $v = v_0+at$. Given $v_0 = 5\ m/s$, $v = 0\ m/s$ and $t = 25\ s$. Rearranging for $a$ gives $a=\frac{v - v_0}{t}$.
$a=\frac{0 - 5}{25}=-0.2\ m/s^{2}$

Problem 5:

Step1: Use the kinematic - equation

We use the equation $d=v_0t+\frac{1}{2}at^{2}$ and $v = v_0+at$. First, from $d = v_0t+\frac{1}{2}at^{2}$, given $v_0 = 14.4\ m/s$, $t = 11\ s$ and $d = 132\ m$. Substitute into the equation: $132=14.4\times11+\frac{1}{2}a\times11^{2}$.
$132 = 158.4+\frac{121a}{2}$. Then $\frac{121a}{2}=132 - 158.4=-26.4$, and $a=\frac{-26.4\times2}{121}\approx - 0.437\ m/s^{2}$

Step2: Calculate the final velocity

Using $v = v_0+at$, substitute $v_0 = 14.4\ m/s$, $a=-0.437\ m/s^{2}$ and $t = 11\ s$.
$v=14.4+( - 0.437)\times11=14.4-4.807 = 9.593\ m/s$

Answer:

Problem 1: $t\approx5.04\ s$
Problem 2: $t = 10\ s$
Problem 3: $v_0 = 16.4\ m/s$
Problem 4: $a=-0.2\ m/s^{2}$
Problem 5: $v\approx9.59\ m/s$