Rate, Distance, and Time Problems, 9

Category: Mechanics, Physics

"Published in Newark, California, USA"

Felix rode 18 miles downstream and back to the same place where he started, in 3 hours and 12 minutes. Find the rate of Felix in still water if the velocity of the stream is 3 miles per hour.

Solution:

To illustrate the problem, it is better to draw the figure as follows 

Photo by Math Principles in Everyday Life

Let s = be the distance traveled by Felix in upstream / downstream
      v = be the velocity or speed of Felix in still water
      v₁ = be the velocity in downstream
      v₂ = be the velocity in upstream 
       t₁ = time traveled in downstream
       t₂ = time traveled in upstream 

If the word statement says "Felix rode 18 miles downstream and back to the same place where he started, in 3 hours and 12 minutes.", then the working equation is

Substitute the values of t₁ and t₂ so that we can solve for the value of v which is Felix' speed or velocity in still water. 

By using quadratic formula, the value of v is 

We need to choose the positive sign not the negative sign because the value of velocity will be negative which is not accepted as a final answer.  Therefore, the velocity or speed of Felix in still water is

 

Projectile Motion Problems

Category: Mechanics, Physics

"Published in Newark, California, USA"

An object falls from a plane flying horizontally at an altitude of 40,000 ft. at 500 mi/hr. How long will it take to hit the ground? What is the horizontal distance traveled of an object?

Solution:

The given problem is about projectile motion in which an object is thrown from its maximum height with an initial velocity. The projectile curve is a half-parabola as shown in the figure below

Photo by Math Principles in Everyday Life

First, let's consider the y-component of an object in order to calculate the time required to hit the ground. From the distance formula
 

Since an object is thrown in a downward position, then the values of s_y and a_y are negative. a_y is also equal to g = 32.174 ft/sec²  which is an acceleration due to gravity. V_0y is zero because the direction of initial velocity of an object is horizontal. Therefore, the time required of an object to hit the ground is
 

          

 

 

 

 

Next, let's consider the x-component of an object in order to calculate the horizontal distance traveled of an object. From the distance formula

Since the direction of v_0x is horizontal and constant, then a_x is equal to zero. Therefore, the horizontal distance traveled of an object is

                      or

Rate, Distance, and Time Problems, 8

Category: Physics, Mechanics

"Published in Newark, California, USA"

A runner A can run the mile race in 4.25 min. Another runner B requires 4.55 min to run this distance. If they start out together and maintain their normal speeds, how far apart will they be at the finish of the race?

Solution:

The given problem is about rate, distance, and time problem in which the two runners run in the same mile race. From the given problem also, runner A is faster than runner B because runner A has a lesser time to complete the mile race than runner B.

The speed of runner A is

The speed of runner B is

If the two runners will start together in a mile race, then runner A will finish the race. Therefore, their distance between the two runners at the end of the race is

For runner B, we use the time for runner A because runner A will finish the race.

                        or