Chapter 13, Problem 13.1CYU

Check Your Understanding What happens to force and accleration as the vehicles fall together? What wil our estimate of the velocity at a collision higher or lower than the speed actually be? And finally, what would happen if the masses were not identical? Would the force on each be the same or different? How about their accelerations?

To determine

The effect on force and acceleration when the vehicles fall together.

The estimate of the velocity at the higher or collision than the actual speed.

The effect on force and acceleration if the masses were not identical.

Answer to Problem 13.1CYU

Gravitation force, F, between two bodies and acceleration, a, due to the force is inversely proportional to the square of the distance which infers the increase in F and a at higher rate due to which speed attained by body will be greater. Change in mass will change the acceleration linearly while force will follow the Newton’s third law, that is equal and opposite.

Explanation of Solution

Introduction:

Gravitational force exerted by one body of mass $m_1$ on another body of mass $m_2$ when both are separated by distance of r, is given below

$F_{12}=-F_{21}=\frac{G{m}_1{m}_2}{r^2}\dots \dots \dots \dots \dots \left(1\right)$

Acceleration a of body of mass $m_1$ is given by the following expression

$a=\frac{F}{m_1}=\frac{\frac{G{m}_1{m}_2}{r^2}}{m_1}=\frac{G{m}_2}{r^2}\dots \dots \dots \dots .\left(2\right)$

• Determine force

From equation 2, it is observed that gravitational force is inversely proportional to square of the distance between the two bodies. Force between two bodies when they are at distance r is given below

$F_{initial}=\frac{G{m}_1{m}_2}{r^2}\dots .\left(3\right)$

If body is moved closer half of the initial distance, that is $R=\frac{r}{2}$ force would be given below

$F_{new}=\frac{G{m}_1{m}_2}{R^2}=\frac{G{m}_1{m}_2}{\left({\frac{r}{2}}^2\right)}=\frac{4G{m}_1{m}_2}{r^2}=4F_{initial}\dots \dots \left(4\right)$

Therefore, force increased to 4 times as distance is decreased.Hence, force increases at a faster rate as the two vehicles brought closer.

• Determine acceleration

Acceleration also follows the similarfashion force, as given by equation 2. The new acceleration

$a=\frac{F}{m_1}=\frac{G{m}_2}{{\left(r/2\right)}^2}=\frac{4\times G{m}_2}{r^2}\dots .\left(5\right)$

• Determine speed

Final speed after moving a distance of scan be used using the following equation

$v_f^2-v_i^2=2as$

As the acceleration increases at higher rate, hence the speed of the body (payload) will also be greater.

• Effect mass on force and acceleration

If the mass is changed (increased or decreased), force would also be changed accordingly. But according to Newton’s third law, gravitational force acting on each other would always be equal and directing opposite. As $F_{12}=-F_{21}$. On the other hand, acceleration will be different if the mass is changed.

Conclusion:

Hence, gravitation force, F, between two bodies and acceleration, a, due to the force is inversely proportional to the square of the distance $\left(F\left(ora\right)\propto \frac{1}{r^2}\right)$ hencedue to which speed attained by the body will be greater. Change in mass will change the acceleration linearly while force will follow the Newton’s third law, that is equal and opposite.