Question

Figure 4-53 shows the straight path of a particle across a x-ycoordinate system as the particle is accelerated from rest during time interval $\mathbf{∆}{\mathbf{t}}_{\mathbf{1}}$. The acceleration is constant. Thecoordinates for point B are $(12.0m,18.0m)$; those for pointare (a) What is the ratio ${\mathbf{a}}_{\mathbf{y}}\mathbf{/}{\mathbf{a}}_{\mathbf{x}}$of the acceleration components? (b) What are the coordinates of the particle if the motion is continued for another interval equal to $\mathbf{∆}{\mathbf{t}}_{\mathbf{1}}$?

Short answer

a) The ratio of the acceleration components is. 1.5.

b) The new x and y coordinates of a particle are$\left(36\mathrm{m},54\mathrm{m}\right)$.

Step-by-step solution

The given data

$$\begin{gathered} 1)\mathrm{Time}\mathrm{interval}\mathrm{is}\mathrm{t}=∆{\mathrm{t}}_1 \\ 2)\mathrm{Initial}\mathrm{velocity},{\mathrm{V}}_{0\mathrm{x}}=0\mathrm{and}{\mathrm{V}}_{\mathrm{oy}}=0 \\ 3)\mathrm{The}\mathrm{old}\mathrm{coordinates}\mathrm{are}:\mathrm{A}=\left(4.00\mathrm{m},6.00\mathrm{m}\right)\mathrm{and}\mathrm{B}\left(12.0\mathrm{m},18.0\mathrm{m}\right) \end{gathered}$$

Understanding the concept of the projectile motion

Projectile motion is the motion of a particle that is launched with an initial velocity. During its flight, the particle’s horizontal acceleration is zero and its vertical acceleration is the free-fall acceleration. (Upward is taken to be a positive direction.)

Using the projectile motion equations for motion along x and motion along the y-direction, we can calculate the ratio of acceleration components.

Formulae:

The second horizontal distance equation due to kinematics motion,

$\mathrm{x}-{\mathrm{x}}_0={\mathrm{V}}_{\mathrm{ox}}\mathrm{t}+\frac{1}{2}{\mathrm{a}}_{\mathrm{x}}{\mathrm{t}}^2$ …(i)

The second vertical distance equation due to kinematics motion,

$\mathrm{y}-{\mathrm{y}}_0={\mathrm{V}}_{0\mathrm{y}}\mathrm{t}+\frac{1}{2}{\mathrm{a}}_{\mathrm{y}}{\mathrm{t}}^2$ …(ii)

(a) Calculation of the ratio of the vertical acceleration to the horizontal acceleration

By using the given values in equation (i), we get the equation as:

$$\begin{gathered} \left(12\mathrm{m}-4\mathrm{m}\right)=0+\frac{1}{2}{\mathrm{a}}_{\mathrm{x}}∆{\mathrm{t}}_1^2 \\ 8\mathrm{m}=\frac{1}{2}{\mathrm{a}}_{\mathrm{x}}∆{\mathrm{t}}_1^2 \end{gathered}$$ …(iii)

Similarly using the same data in equation (ii), we get the equation as:

$$\begin{gathered} \left(18\mathrm{m}-6\mathrm{m}\right)=0+\frac{1}{2}{\mathrm{a}}_{\mathrm{y}}∆{\mathrm{t}}_1^2 \\ 12\mathrm{m}=\frac{1}{2}{\mathrm{a}}_{\mathrm{y}}∆{\mathrm{t}}_1^2 \end{gathered}$$ …(iv)

Now, dividing eq. (iv) by eq. (iii), we get the required value of the ratio of the vertical to the horizontal accelerations as follows:

$$\begin{gathered} \frac{{\mathrm{a}}_{\mathrm{y}}}{{\mathrm{a}}_{\mathrm{x}}}=\frac{12.0\mathrm{m}}{8.00\mathrm{m}} \\ =1.5 \end{gathered}$$

Hence, the value of the ratio is 1.5.

(b) Calculation of the new coordinates of the particle

It is given that the motion continues for additional time interval which is same as

For this part, we have the time interval as:$∆{\mathrm{t}}_2=2∆{\mathrm{t}}_1$

Substituting the given values in equation (i) we have

$8m=\frac{1}{2}{a}_x∆t_1^2$

Now for new displacement we can write the same equation (i) as follows:

$∆x=\frac{1}{2}{a}_x∆t_2^2$

Taking the ratio of these two above equations, we get the-coordinate as:

role="math" localid="1657017179060" $$\begin{gathered} \frac{∆x}{8m}=\frac{∆t_2^2}{∆t_1^2} \\ ∆x=\left(8.00m\right)\times 2^2\left(\because ∆t_2=2∆t_1\right) \\ =32m \end{gathered}$$

Similarly, using the given data in equation (ii), we get the-coordinate as:

$$\begin{gathered} ∆\mathrm{y}=\left(12.0\mathrm{m}\right)\times 2^2 \\ =48\mathrm{m} \end{gathered}$$

So, the new co-ordinates will be, (considering the previous distance)

$$\begin{gathered} \mathrm{x}=32\mathrm{m}+4\mathrm{m} \\ =36\mathrm{m} \\ \mathrm{y}=48\mathrm{m}+6\mathrm{m} \\ =54\mathrm{m} \end{gathered}$$

Hence, the value of the new coordinates of the particle is$\left(36\mathrm{m},54\mathrm{m}\right)$