Question

problem 1. a spanish ball is accelerating at a rate of 2.6 m/s² until it reaches a final velocity of 10.5 m/s. if the balls displacement is 20 m, calculate the initial velocity.
problem 2. a dog is accelerating at a rate of 1.6 m/s² until it reaches a final velocity of 20 m/s. if the dogs displacement is 25 m, calculate the dogs initial velocity.
problem 3. angelique is moving with an initial velocity of 4 m/s. she accelerates at a rate of 1.5 m/s² until reaching a final velocity of 10 m/s. calculate angeliques displacement.
problem 4. a runner accelerates from rest at a rate of 3 m/s². determine the amount of time it would take the runner to travel a displacement of 108 m.

Explanation:

Step1: Identify the kinematic - equation

We use the equation $v_f^2=v_i^2 + 2ad$, where $v_f$ is the final velocity, $v_i$ is the initial velocity, $a$ is the acceleration and $d$ is the displacement.

Problem 1:

Given $a = 2.6\ m/s^2$, $v_f=10.5\ m/s$, $d = 20\ m$.

Step1: Rearrange the kinematic - equation for $v_i$

$v_i^2=v_f^2 - 2ad$
$v_i=\sqrt{v_f^2 - 2ad}$

Step2: Substitute the values

$v_i=\sqrt{(10.5)^2-2\times2.6\times20}$
$v_i=\sqrt{110.25 - 104}$
$v_i=\sqrt{6.25}$
$v_i = 2.5\ m/s$

Problem 2:

Given $a = 1.6\ m/s^2$, $v_f = 20\ m/s$, $d = 25\ m$.

Step1: Rearrange the kinematic - equation for $v_i$

$v_i^2=v_f^2 - 2ad$
$v_i=\sqrt{v_f^2 - 2ad}$

Step2: Substitute the values

$v_i=\sqrt{(20)^2-2\times1.6\times25}$
$v_i=\sqrt{400 - 80}$
$v_i=\sqrt{320}$
$v_i\approx17.9\ m/s$

Problem 3:

Given $v_i = 4\ m/s$, $v_f = 10\ m/s$, $a = 1.5\ m/s^2$.

Step1: Rearrange the kinematic - equation for $d$

$v_f^2=v_i^2 + 2ad$
$d=\frac{v_f^2 - v_i^2}{2a}$

Step2: Substitute the values

$d=\frac{(10)^2-(4)^2}{2\times1.5}$
$d=\frac{100 - 16}{3}$
$d=\frac{84}{3}$
$d = 28\ m$

Problem 4:

Given $v_i = 0\ m/s$ (starts from rest), $a = 3\ m/s^2$, $d = 108\ m$.
We use the equation $d=v_it+\frac{1}{2}at^2$. Since $v_i = 0$, the equation simplifies to $d=\frac{1}{2}at^2$.

Step1: Rearrange the equation for $t$

$t=\sqrt{\frac{2d}{a}}$

Step2: Substitute the values

$t=\sqrt{\frac{2\times108}{3}}$
$t=\sqrt{72}$
$t\approx8.5\ s$

Answer:

Problem 1: $2.5\ m/s$
Problem 2: $\approx17.9\ m/s$
Problem 3: $28\ m$
Problem 4: $\approx8.5\ s$