Speed of Truck Relative to Highway: Solving the Puzzle

[kent davidge]

Homework Statement

You are in your car driving on a highway at 25 m/s when you glance in the passenger-side mirror (a convex mirror with radius of curvature 150 cm) and notice a truck approaching. If the image of the truck is approaching the vertex of the mirror at a speed of when the truck is 2.0 m from the mirror, what is the speed of the truck relative to the highway?

Homework Equations

The Attempt at a Solution

(sorry my bad english)
Assuming both the truck and the car speeds to be constant, the truck takes the same time interval as its image to reach the vertex of the mirror. So,

1 / f = 1 / s + 1 / s'
-1 / 0.75 m = 1 / 2 m + 1 / s'
s' = -0.54 m
t = -0.54 m / -1.9 m /s
[ (V - 25) m / s ] t = 2 m
but solving this for V I didnt find the correct value: 51 m / s. Maybe the speed is'nt constant?

My sketch:

 

[haruspex]

The speed of the image will not be constant. You can see this by differentiating the 1/f formula.

 

[kent davidge]

I don't know how to do this

 

[haruspex]

I don't know how to do this

The derivative of s with respect to time, ds/dt, is the velocity, v. What is the derivative of 1/s? Use the chain rule.

 

[kent davidge]

hmmmm let me try...

1 / f = 1 / V dt + 1 / V' dt
1 / f = 1 / dt (1 / V + 1 / V')

but how can I solve this derivative? would I need to express f as function of t?

 

[haruspex]

hmmmm let me try...

1 / f = 1 / V dt + 1 / V' dt
1 / f = 1 / dt (1 / V + 1 / V')

but how can I solve this derivative? would I need to express f as function of t?

f is a constant. What is the derivative of a constant?
The chain rule says d/dt(1/s)= (ds/dt)(d/ds(1/s)). What is d/ds(1/s)?

 

[kent davidge]

ohh yah

(d / dt) 1 / f = 0
0 = - [(V - 25 m / s) / s²] - V' / s'²
V ≅ 51 m / s

Thanks

but why the image speed is not constant?

 

[haruspex]

ohh yah

(d / dt) 1 / f = 0
0 = - [(V - 25 m / s) / s²] - V' / s'²
V ≅ 51 m / s

Thanks

but why the image speed is not constant?

Because the relationship between the two distances is nonlinear.