Chapter 6.3, Problem 39PPS

(a)

To determine

Solve the equation for v.

Answer to Problem 39PPS

  $v=\sqrt{\frac{E}{0.5m}}$

Explanation of Solution

Given:

The formula for kinetic energy , measured in joules j , is $E=0.5m{v}^2$ , where m is the mass in kilograms and v is the velocity of the object in meters per second.

Calculation:

Given:

  $E=0.5m{v}^2$

Solve it for v:

  $\begin{array}{l}E=0.5m{v}^2\\ \Rightarrow v^2=\frac{E}{0.5m}\mathrm{...........}[\text{divide each side by 0}\text{.5m]}\\ \Rightarrow v=\pm \sqrt{\frac{E}{0.5m}}\mathrm{.......}[\text{take square root each side]}\end{array}$

Since v is the velocity , it cannot be negative .

So,

  $v=\sqrt{\frac{E}{0.5m}}$

(b)

To determine

If a 1500-kilogram vehicle is generating 1 million joules of kinetic energy , find its speed.

Answer to Problem 39PPS

velocity is about 36.5 m/s

Explanation of Solution

Given:

The formula for kinetic energy , measured in joules j , is $E=0.5m{v}^2$ , where m is the mass in kilograms and v is the velocity of the object in meters per second.

Calculation:

From part (a), we have:

  $v=\sqrt{\frac{E}{0.5m}}$

Substitute :

m = 1500 kilograms

E = 1 million joules = ${10}^6$ joules$\begin{array}{l}v=\sqrt{\frac{E}{0.5m}}\\ v=\sqrt{\frac{{10}^6}{0.5(1500)}}\\ v=\sqrt{\frac{{10}^6\cdot 10}{5\cdot 15\cdot {10}^2}}\\ v=\sqrt{\frac{{10}^{6+1-2}}{75}}\mathrm{............}[a^{x+y}=a^x\cdot a^y\&a^{x-y}=\frac{a^x}{a^y}]\\ v=\sqrt{\frac{{10}^5}{75}}\\ v=\sqrt{1333.3333}\\ v=36.514836\\ v\approx 36.5m/s\end{array}$

So, the velocity is about 36.5 m/s.

(c)

To determine

Estimate the escape velocity of Jupiter.

Answer to Problem 39PPS

The escape velocity of Jupiter is about 85,100 m/s.

Explanation of Solution

Given:

Suppose a 100,000 kilogram ship must have a kinetic energy of $3.624\times {10}^{14}$ joules to escape the gravitational field of Jupiter.

Calculation:

From part (a), we have:

  $v=\sqrt{\frac{E}{0.5m}}$

Substitute :

m = 100,000 kilograms

E = $3.624\times {10}^{14}$ joules$\begin{array}{l}v=\sqrt{\frac{E}{0.5m}}\\ v=\sqrt{\frac{3.624\times {10}^{14}}{0.5(100,000)}}\\ v=\sqrt{\frac{2\times 3.624\times {10}^{14}}{{10}^5}}\\ v=\sqrt{7.248\times {10}^{14-5}}\mathrm{............}[a^{x+y}=a^x\cdot a^y\&a^{x-y}=\frac{a^x}{a^y}]\\ v=\sqrt{7.248\times {10}^9}\\ v=\sqrt{72.48\times {10}^8}\\ v=8.513518662\times {10}^4\\ v\approx 85,135m/s\end{array}$

So, the escape velocity of Jupiter is about 85,100 m/s.