<< Chapter < Page Chapter >> Page >

Making connections: force and momentum

Force and momentum are intimately related. Force acting over time can change momentum, and Newton’s second law of motion, can be stated in its most broadly applicable form in terms of momentum. Momentum continues to be a key concept in the study of atomic and subatomic particles in quantum mechanics.

This statement of Newton’s second law of motion includes the more familiar F net = m a as a special case. We can derive this form as follows. First, note that the change in momentum Δ p size 12{Δp} {} is given by

Δ p = Δ ( m v ) . size 12{Δp=Δ left (mv right )} {}

If the mass of the system is constant, then

Δ ( m v ) = m Δ v . size 12{Δ left (mv right )=mΔv} {}

So that for constant mass, Newton’s second law of motion becomes

F net = Δ p Δ t = m Δ v Δ t . size 12{ F rSub { size 8{"net"} } = { {Δp} over {Δt} } = { {mΔv} over {Δt} } "." } {}

Because Δ v Δ t = a size 12{ { {Δv} over {Δt} } =a} {} , we get the familiar equation

F net = m a

when the mass of the system is constant .

Newton’s second law of motion stated in terms of momentum is more generally applicable because it can be applied to systems where the mass is changing, such as rockets, as well as to systems of constant mass. We will consider systems with varying mass in some detail ; however, the relationship between momentum and force remains useful when mass is constant, such as in the following example.

Calculating force: venus williams’ racquet

During the 2007 French Open, Venus Williams hit the fastest recorded serve in a premier women’s match, reaching a speed of 58 m/s (209 km/h). What is the average force exerted on the 0.057-kg tennis ball by Venus Williams’ racquet, assuming that the ball’s speed just after impact is 58 m/s, that the initial horizontal component of the velocity before impact is negligible, and that the ball remained in contact with the racquet for 5.0 ms (milliseconds)?

Strategy

This problem involves only one dimension because the ball starts from having no horizontal velocity component before impact. Newton’s second law stated in terms of momentum is then written as

F net = Δ p Δ t . size 12{ F rSub { size 8{"net"} } = { {Δp} over {Δt} } = { {mΔv} over {Δt} } "." } {}

As noted above, when mass is constant, the change in momentum is given by

Δ p = m Δ v = m v f v i . size 12{Δp=mΔv=m left (v rSub { size 8{f} } - v rSub { size 8{i} } right )} {}

In this example, the velocity just after impact and the change in time are given; thus, once Δ p size 12{Δp} {} is calculated, F net = Δ p Δ t size 12{ F rSub { size 8{"net"} } = { {Δp} over {Δt} } } {} can be used to find the force.

Solution

To determine the change in momentum, substitute the values for the initial and final velocities into the equation above.

Δ p = m v f v i = 0.057 kg 58 m/s 0 m/s = 3 .306 kg · m/s 3.3 kg · m/s

Now the magnitude of the net external force can determined by using F net = Δ p Δ t size 12{ F rSub { size 8{"net"} } = { {Δp} over {Δt} } } {} :

F net = Δ p Δ t = 3.306 kg m/s 5 . 0 × 10 3 s = 661 N 660 N, alignl { stack { size 12{ F rSub { size 8{"net"} } = { {Δp} over {Δt} } = { {3 "." "306"`"kg" cdot "m/s"} over {5 "." 0 times "10" rSup { size 8{ - 3} } `s} } } {} #" "="661 N" approx "660"`"N," {} } } {}

where we have retained only two significant figures in the final step.

Discussion

This quantity was the average force exerted by Venus Williams’ racquet on the tennis ball during its brief impact (note that the ball also experienced the 0.56-N force of gravity, but that force was not due to the racquet). This problem could also be solved by first finding the acceleration and then using F net = ma size 12{F rSub { size 8{"net"} } " = " ital "ma"} {} , but one additional step would be required compared with the strategy used in this example.

Got questions? Get instant answers now!

Section summary

  • Linear momentum ( momentum for brevity) is defined as the product of a system’s mass multiplied by its velocity.
  • In symbols, linear momentum p is defined to be
    p = m v , size 12{p=mv} {}
    where m size 12{m} {} is the mass of the system and v size 12{v} {} is its velocity.
  • The SI unit for momentum is kg · m/s size 12{"kg" cdot "m/s"} {} .
  • Newton’s second law of motion in terms of momentum states that the net external force equals the change in momentum of a system divided by the time over which it changes.
  • In symbols, Newton’s second law of motion is defined to be
    F net = Δ p Δ t , size 12{ F rSub { size 8{"net"} } = { {Δp} over {Δt} } = { {mΔv} over {Δt} } "." } {}
    F net is the net external force, Δ p size 12{Δp} {} is the change in momentum, and Δ t size 12{Δt} {} is the change time.

Conceptual questions

An object that has a small mass and an object that has a large mass have the same momentum. Which object has the largest kinetic energy?

Got questions? Get instant answers now!

An object that has a small mass and an object that has a large mass have the same kinetic energy. Which mass has the largest momentum?

Got questions? Get instant answers now!

Professional Application

Football coaches advise players to block, hit, and tackle with their feet on the ground rather than by leaping through the air. Using the concepts of momentum, work, and energy, explain how a football player can be more effective with his feet on the ground.

Got questions? Get instant answers now!

How can a small force impart the same momentum to an object as a large force?

Got questions? Get instant answers now!

Problems&Exercises

(a) Calculate the momentum of a 2000-kg elephant charging a hunter at a speed of 7 . 50 m/s size 12{7 "." "50"``"m/s"} {} . (b) Compare the elephant’s momentum with the momentum of a 0.0400-kg tranquilizer dart fired at a speed of 600 m/s size 12{"600"``"m/s"} {} . (c) What is the momentum of the 90.0-kg hunter running at 7 . 40 m/s size 12{7 "." "40"``"m/s"} {} after missing the elephant?

(a) 1.50 × 10 4 kg m/s size 12{1 "." "50" times "10" rSup { size 8{4} } `"kg" cdot "m/s"} {}

(b) 625 to 1

(c) 6 . 66 × 10 2 kg m/s size 12{6 "." "66" times "10" rSup { size 8{2} } `"kg" cdot "m/s"} {}

Got questions? Get instant answers now!

(a) What is the mass of a large ship that has a momentum of 1 . 60 × 10 9 kg · m/s , when the ship is moving at a speed of 48.0 km/h? size 12{"48" "." 0``"km/h?"} {} (b) Compare the ship’s momentum to the momentum of a 1100-kg artillery shell fired at a speed of 1200 m/s size 12{"1200"``"m/s"} {} .

Got questions? Get instant answers now!

(a) At what speed would a 2 . 00 × 10 4 -kg size 12{2 "." "00" times "10" rSup { size 8{4} } "-kg"} {} airplane have to fly to have a momentum of 1 . 60 × 10 9 kg · m/s size 12{1 "." "60" times "10" rSup { size 8{9} } "kg" cdot "m/s"} {} (the same as the ship’s momentum in the problem above)? (b) What is the plane’s momentum when it is taking off at a speed of 60.0 m/s size 12{"60" "." 0``"m/s"} {} ? (c) If the ship is an aircraft carrier that launches these airplanes with a catapult, discuss the implications of your answer to (b) as it relates to recoil effects of the catapult on the ship.

(a) 8 . 00 × 10 4 m/s size 12{8 "." "00" times "10" rSup { size 8{4} } " m/s"} {}

(b) 1 . 20 × 10 6 kg · m/s size 12{1 "." "20" times "10" rSup { size 8{6} } " kg" cdot "m/s"} {}

(c) Because the momentum of the airplane is 3 orders of magnitude smaller than of the ship, the ship will not recoil very much. The recoil would be 0 . 0100 m/s size 12{ - 0 "." "0100"`"m/s"} {} , which is probably not noticeable.

Got questions? Get instant answers now!

(a) What is the momentum of a garbage truck that is 1 . 20 × 10 4 kg size 12{1 "." "20" times "10" rSup { size 8{4} } " kg"} {} and is moving at 10 . 0 m/s size 12{10 "." "0 m/s"} {} ? (b) At what speed would an 8.00-kg trash can have the same momentum as the truck?

Got questions? Get instant answers now!

A runaway train car that has a mass of 15,000 kg travels at a speed of 5 .4 m/s size 12{5 "." 4`"m/s"} {} down a track. Compute the time required for a force of 1500 N to bring the car to rest.

54 s

Got questions? Get instant answers now!

The mass of Earth is 5 . 972 × 10 24 kg size 12{5 "." "972" times 10 rSup { size 8{"24"} } " kg"} {} and its orbital radius is an average of 1 . 496 × 10 11 m size 12{1 "." "496" times 10 rSup { size 8{"11"} } " m"} {} . Calculate its linear momentum.

Got questions? Get instant answers now!

Questions & Answers

draw magnetic field pattern for two wire carrying current in the same direction
Ven Reply
An American traveler in New Zealand carries a transformer to convert New Zealand’s standard 240 V to 120 V so that she can use some small appliances on her trip.
nkombo Reply
What is the ratio of turns in the primary and secondary coils of her transformer?
nkombo
How electric lines and equipotential surface are mutually perpendicular?
Abid Reply
The potential difference between any two points on the surface is zero that implies È.Ŕ=0, Where R is the distance between two different points &E= Electric field intensity. From which we have cos þ =0, where þ is the angle between the directions of field and distance line, as E andR are zero. Thus
MAHADEV
sorry..E and R are non zero...
MAHADEV
By how much leeway (both percentage and mass) would you have in the selection of the mass of the object in the previous problem if you did not wish the new period to be greater than 2.01 s or less than 1.99 s?
Elene Reply
what Is linear momentum
Victoria Reply
why no diagrams
Blessing Reply
where
Fayyaz
Myanmar
Pyae
hi
Iroko
hello
Abdu
Describe an experiment to determine short half life
Tyson Reply
what is science
Kenedy Reply
it's a natural phenomena
Hassan
sap
Emmanuel
please can someone help me with explanations of wave
Benedine
A 20MH coil has a resistance of 50 ohms and us connected in series with a capacitor to a 520MV supply
Musa Reply
what is physics
Caya Reply
it is the science which we used in our daily life
Sujitha
Physics is the branch of science that deals with the study of matter and the interactions it undergoes with energy
Junior
it is branch of science which deals with study of happening in the human life
AMIT
A 20MH coil has a resistance of 50 ohms and is connected in series with a capacitor to a 250MV supply if the circuit is to resonate at 100KHZ, Determine 1: the capacitance of the capacitor 2: the working voltage of the circuit, given that pie =3.142
Musa
Physics is the branch of science that deals with the study of matter and the interactions it undergoes with energy
Kelly
Heat is transfered by thermal contact but if it is transfered by conduction or radiation, is it possible to reach in thermal equilibrium?
Eden Reply
Yes, It is possible by conduction if Surface is Adiabatic
Astronomy
Yeah true ilwith d help of Adiabatic
Kelly
what are the fundamentals qualities
Magret Reply
what is physic3
Kalilu
what is physic
Kalilu
Physics? Is a branch of science dealing with matter in relation to energy.
Moses
Physic... Is a purging medicine, which stimulates evacuation of the bowels.
Moses
are you asking for qualities or quantities?
Noman
give examples of three dimensional frame of reference
Ekwunazor Reply
Universe
Noman
Yes the Universe itself
Astronomy
Examine different types of shoes, including sports shoes and thongs. In terms of physics, why are the bottom surfaces designed as they are? What differences will dry and wet conditions make for these surfaces?
Lathan Reply
sports shoes are designed in such a way they are gripped well with your feet and their bases have and high friction surfaces, Thong shoes are for comfort, these are easily removed and light weight. these are usually low friction surfaces but in wet conditions they offer greater friction.
Noman
thong sleepers are usually used in restrooms.
Noman
what is wave
Ochigbo Reply
The phenomenon of transfer of energy
Noman
Practice Key Terms 2

Get the best College physics course in your pocket!





Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'College physics' conversation and receive update notifications?

Ask