Chapter 2, Problem 68GP

(a)

To determine

The time taken by fugitive to catch up to the empty box car

Answer to Problem 68GP

4 seconds.

Explanation of Solution

Given:

Initial velocity of the fugitive $=v_o=0\frac{\text{m}}{\text{s}}$

Final maximum speed of the fugitive $=v_{\max }=8\frac{\text{m}}{\text{s}}$

Acceleration of the fugitive $=a=4\frac{\text{m}}{{\text{s}}^{\text{2}}}$

Time taken by the fugitive to accelerate to the maximum speed $=t$

Displacement of the fugitive while accelerating $=x$

Time spent by the fugitive to travel at maximum speed $=t_{\max }$

Total time of travel for fugitive to catch the car $=t_{tot}$

Constant speed of the car $=v_{car}=6\frac{\text{m}}{\text{s}}$

Total displacement of the car $=x_{car}$

Formula Used:

Acceleration is given as

  $a=\frac{v_f-v_o}{t}$

Displacement is given as

  $x=v_{o}t+(0.5)a{t}^2$

Displacement at constant velocity is given as

  $\text{displacement = (constant velocity)(time)}$

Calculation:

Consider the motion of the fugitive while speeding up:

Average acceleration of the fugitive is given as

  $a=\frac{v_{\max }-v_o}{t}$

Inserting the values

  $\begin{array}{l}4=\frac{8-0}{t}\\ t=2\sec \end{array}$

Displacement of the fugitive while accelerating is given as

  $x=v_{o}t+(0.5)a{t}^2$

Substituting the values

  $\begin{array}{l}x=(0)(2)+(0.5)(4){(}^2\\ x=8\text{m}\end{array}$

For the fugitive to catch the car,

Total displacement of fugitive = Total displacement of car

Displacement of fugitive while accelerating + displacement of fugitive at maximum speed = Total displacement of car

  $\begin{array}{l}x+x_{\max }=x_{car}\\ 8+v_{\max }{t}_{\max }=v_{car}{t}_{tot}\\ 8+(8)t_{\max }=v_{car}(t+t_{\max })\\ 8+(8)t_{\max }=(6)(2+t_{\max })\\ 8+8t_{\max }=12+6t_{\max }\\ t_{\max }=2\sec \end{array}$

So total time to catch the car is given as

  $\begin{array}{l}{t}_{tot}=t+t_{\max }\\ t_{tot}=2+2\\ t_{tot}=4\sec \end{array}$

Conclusion:

Therefore, time taken by fugitive is 4 seconds.

(b)

To determine

The distance travelled by fugitive to reach the box car.

Answer to Problem 68GP

24 m

Explanation of Solution

Given:

Total time of travel for fugitive to catch the car $=t_{tot}=4\sec$

Constant speed of the car $=v_{car}=6\frac{\text{m}}{\text{s}}$

Total distance travelled by fugitive $=x_{fugitive}$

Total distance travelled by car $=x_{car}$

Formula Used:

Distance travelled at constant speed is given as

  $\text{distance = (constant speed)(time)}$

Calculation:

For the fugitive to catch the box car,

Total distance travelled by fugitive = total distance travelled by car

  $\begin{array}{l}{x}_{fugitive}=x_{car}\\ x_{fugitive}=v_{car}{t}_{tot}\\ x_{fugitive}=(6)(4)\\ x_{fugitive}=24\text{m}\end{array}$

Conclusion:

Therefore, distance travelled by fugitive is 24 m.