Apply problem-solving techniques to solve for quantities in more complex systems of forces.
Integrate concepts from kinematics to solve problems using Newton's laws of motion.
There are many interesting applications of Newton’s laws of motion, a few more of which are presented in this section. These serve also to illustrate some further subtleties of physics and to help build problem-solving skills.
Drag force on a barge
Suppose two tugboats push on a barge at different angles, as shown in
[link] . The first tugboat exerts a force of
in the
x -direction, and the second tugboat exerts a force of
in the
y -direction.
If the mass of the barge is
and its acceleration is observed to be
in the direction shown, what is the drag force of the water on the barge resisting the motion? (Note: drag force is a frictional force exerted by fluids, such as air or water. The drag force opposes the motion of the object.)
Strategy
The directions and magnitudes of acceleration and the applied forces are given in
[link](a) . We will define the total force of the tugboats on the barge as
so that:
Since the barge is flat bottomed, the drag of the water
will be in the direction opposite to
, as shown in the free-body diagram in
[link] (b). The system of interest here is the barge, since the forces on
it
are given as well as its acceleration. Our strategy is to find the magnitude and direction of the net applied force
, and then apply Newton’s second law to solve for the drag force
.
Solution
Since
and
are perpendicular, the magnitude and direction of
are easily found. First, the resultant magnitude is given by the Pythagorean theorem:
The angle is given by
which we know, because of Newton’s first law, is the same direction as the acceleration.
is in the opposite direction of
, since it acts to slow down the acceleration. Therefore, the net external force is in the same direction as
, but its magnitude is slightly less than
. The problem is now one-dimensional. From
[link](b) , we can see that
But Newton’s second law states that
Thus,
This can be solved for the magnitude of the drag force of the water
in terms of known quantities:
Substituting known values gives
The direction of
has already been determined to be in the direction opposite to
, or at an angle of
south of west.
Wayne and Dennis like to ride the bike path from Riverside Park to the beach. Dennis’s speed is seven miles per hour faster than Wayne’s speed, so it takes Wayne 2 hours to ride to the beach while it takes Dennis 1.5 hours for the ride. Find the speed of both bikers.
from theory: distance [miles] = speed [mph] × time [hours]
info #1
speed_Dennis × 1.5 = speed_Wayne × 2
=> speed_Wayne = 0.75 × speed_Dennis (i)
info #2
speed_Dennis = speed_Wayne + 7 [mph] (ii)
use (i) in (ii) => [...]
speed_Dennis = 28 mph
speed_Wayne = 21 mph
George
Let W be Wayne's speed in miles per hour and D be Dennis's speed in miles per hour. We know that W + 7 = D and W * 2 = D * 1.5.
Substituting the first equation into the second:
W * 2 = (W + 7) * 1.5
W * 2 = W * 1.5 + 7 * 1.5
0.5 * W = 7 * 1.5
W = 7 * 3 or 21
W is 21
D = W + 7
D = 21 + 7
D = 28
Salma
Devon is 32 32 years older than his son, Milan. The sum of both their ages is 54 54. Using the variables d d and m m to represent the ages of Devon and Milan, respectively, write a system of equations to describe this situation. Enter the equations below, separated by a comma.
please why is it that the 0is in the place of ten thousand
Grace
Send the example to me here and let me see
Stephen
A meditation garden is in the shape of a right triangle, with one leg 7 feet. The length of the hypotenuse is one more than the length of one of the other legs. Find the lengths of the hypotenuse and the other leg
however, may I ask you some questions about Algarba?
Amoon
hi
Enock
what the last part of the problem mean?
Roger
The Jones family took a 15 mile canoe ride down the Indian River in three hours. After lunch, the return trip back up the river took five hours. Find the rate, in mph, of the canoe in still water and the rate of the current.
Shakir works at a computer store. His weekly pay will be either a fixed amount, $925, or $500 plus 12% of his total sales. How much should his total sales be for his variable pay option to exceed the fixed amount of $925.
I'm guessing, but it's somewhere around $4335.00 I think
Lewis
12% of sales will need to exceed 925 - 500, or 425 to exceed fixed amount option. What amount of sales does that equal? 425 ÷ (12÷100) = 3541.67. So the answer is sales greater than 3541.67.
Check:
Sales = 3542
Commission 12%=425.04
Pay = 500 + 425.04 = 925.04.
925.04 > 925.00
Munster
difference between rational and irrational numbers
Jazmine trained for 3 hours on Saturday. She ran 8 miles and then biked 24 miles. Her biking speed is 4 mph faster than her running speed. What is her running speed?