Question

problem 1. a spanish bull is accelerating at a rate of 2.5 m/s² until it reaches a final velocity of 17.5 m/s. if the bulls displacement is 20 m, calculate the initial velocity.
problem 2. a dog is accelerating at a rate of 1.5 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 0.7 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 will take the runner to travel a displacement of 108 m.

Explanation:

Problem 1

Step1: Rearrange the formula

We start with $v_f^{2}=v_0^{2}+2a\Delta x$. Rearranging for $v_0$, we get $v_0^{2}=v_f^{2}-2a\Delta x$.

Step2: Substitute values

Given $v_f = 17.5\ m/s$, $a = 2.5\ m/s^{2}$, and $\Delta x=20\ m$. Substitute into the formula: $v_0^{2}=(17.5)^{2}-2\times2.5\times20$.
$v_0^{2}=306.25 - 100=206.25$.

Step3: Solve for $v_0$

Take the square - root of both sides. Since velocity is a physical quantity, we consider the real - valued root. $v_0=\sqrt{206.25}=14.36\ m/s$.

Step1: Rearrange the formula

Starting with $v_f^{2}=v_0^{2}+2a\Delta x$, we rearrange to $v_0^{2}=v_f^{2}-2a\Delta x$.

Step2: Substitute values

Given $v_f = 20\ m/s$, $a = 1.5\ m/s^{2}$, and $\Delta x = 25\ m$. Then $v_0^{2}=(20)^{2}-2\times1.5\times25$.
$v_0^{2}=400 - 75 = 325$.

Step3: Solve for $v_0$

Take the square - root of both sides. $v_0=\sqrt{325}\approx18.03\ m/s$.

Step1: Rearrange the formula

We use $v_f^{2}=v_0^{2}+2a\Delta x$. Rearranging for $\Delta x$, we get $\Delta x=\frac{v_f^{2}-v_0^{2}}{2a}$.

Step2: Substitute values

Given $v_0 = 4\ m/s$, $v_f = 10\ m/s$, and $a = 0.7\ m/s^{2}$. Substitute into the formula: $\Delta x=\frac{(10)^{2}-(4)^{2}}{2\times0.7}$.
$\Delta x=\frac{100 - 16}{1.4}=\frac{84}{1.4}=60\ m$.

Answer:

$14.36\ m/s$

Problem 2