Question

Marc is in no way athletic, never goes to the gym, and gets winded by just walking up some stairs. When he was in his undergrad, Marc used to be late for the bus frequently. One time, he was running at 5 m/s to catch a bus that he was late for while the bus was 40 m away. Marc then noticed the bus started to drive away with a constant acceleration of a = 0.17 m/s^2 (Marc can notice these things).

1. (a) How long does Marc have to continuing running at 5 m/s to catch the bus? How much distance has he covered in that time?

2. (b) When Marc reaches the bus, how fast is the bus traveling?

3. (c) You may have noticed that the mathematical process in part (a) produces two solutions. Explain the physical meaning of the second solution. How fast is the bus traveling at this point?

4. (d) Now, it’s a miracle Marc was able to even achieve a speed of 5 m/s. In all honesty, he can likely only reach 3.5 m/s (at best). At this speed, would Marc catch the bus?

5. (e) What is the minimum speed someone would need to be able to run to catch the bus under these conditions? What would be their running time be and total distance traveled?

Answer 1

a) Marc is running at a speed

Acceleration of bus

Distance between Marc and bus

In time , but starting from rest travels a distance

In the same time , Marc travels distance and catches the bus,

, so

Solving above quadratic equation, and

First time when Marc catches bus is

Hence ,Time taken for Marc to catch bus is

Distance covered by him in this time is

b)

When Marc reaches bus, bus is traveling with a speed .

c)

The second solution to part a) is

The first time Marc reaches the bus in time , then if he continues to run with same speed goes a small distance ahead of bus. Bus with its constant acceleration, again crosses the Marc at time and moves ahead of Marc.

Speed of bust with this time is

d)

Maximum speed of Marc is

With this speed , if he catches bus ,

,

For the above quadratic equation , real solutions are not possible.

Hence Marc cannot catch the bus.

e)

If Marc is traveling with speed   , , distance covered by bus is

To get real solutions, that is

Minimum speed required to catch the bus is