Define linear momentum.Explain the relationship in between momentum and also force.State Newton’s second law of motion in terms of momentum.Calculate momentum provided mass and velocity.

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Linear Momentum

The scientific meaning of linear momentum is continuous with many people’s intuitive understanding of momentum: a large, fast-moving thing has greater momentum than a smaller, slower object. Linear momentum is characterized as the product of a system’s mass multiplied by its velocity. In symbols, straight momentum is expressed as pmv.

Momentum is directly proportional come the object’s mass and likewise its velocity. Hence the greater an object’s fixed or the greater its velocity, the higher its momentum. Momentum p is a vector having the exact same direction together the velocity v. The SI unit because that momentum is kg · m/s.

Linear Momentum

Linear inert is identified as the product of a system’s mass multiply by its velocity:


Example 1. Calculating Momentum: A football Player and a Football

Calculate the inert of a 110-kg football player running at 8.00 m/s.Compare the player’s momentum v the inert of a hard-thrown 0.410-kg football that has actually a rate of 25.0 m/s.Strategy

No info is given concerning direction, and also so we have the right to calculate only the size of the momentum, p. (As usual, a symbol that is in italics is a magnitude, whereas one that is italicized, boldfaced, and also has an arrowhead is a vector.) In both components of this example, the magnitude of momentum can be calculated directly from the meaning of momentum provided in the equation, which becomes p = mv when just magnitudes space considered.

Solution for part 1

To recognize the momentum of the player, instead of the recognized values for the player’s mass and also speed right into the equation.

pplayer = (110 kg)( 8.00 m/s) = 880 kg · m/s

Solution for Part 2

To determine the momentum of the ball, instead of the well-known values because that the ball’s mass and speed right into the equation.

p sphere = (0.410 kg)(25.0 m/s) = 10.3 kg · m/s

The ratio of the player’s inert to that of the round is

displaystylefracp_ extplayerp_ extball=frac88010.3=85.9\


Although the round has higher velocity, the player has a much greater mass. Thus the momentum of the player is much greater than the inert of the football, as you could guess. Together a result, the player’s motion is just slightly impacted if he records the ball. Us shall quantify what wake up in such collisions in terms of momentum in later on sections.

 Momentum and Newton’s second Law

The prestige of momentum, unequal the importance of energy, was recognized at an early stage in the advance of classical physics. Momentum was considered so important that that was referred to as the “quantity of motion.” Newton actually stated his second law of motion in terms of momentum: The net exterior force equates to the adjust in momentum of a system separated by the time over which it changes. Making use of symbols, this legislation is

displaystylemathbfF_ extnet=fracDeltamathbfpDelta t,

where Fnet is the net external force, Δp is the readjust in momentum, and Δt is the readjust in time.

Newton’s second Law of activity in regards to Momentum

The net exterior force equates to the change in momentum of a system divided by the time over which that changes.

displaystylemathbfF_ extnet=fracDeltamathbfpDelta t

Making Connections: Force and also Momentum

Force and also momentum room intimately related. Force acting in time can adjust momentum, and also Newton’s second law the motion, deserve to be stated in that is most extensively applicable form in regards to momentum. Momentum continues to it is in a crucial concept in the study of atomic and subatomic particles in quantum mechanics.

This explain of Newton’s 2nd law the motion consists of the much more familiar Fnet = ma as a special case. We have the right to derive this kind as follows. First, note that the readjust in inert Δp is given by Δp = Δ(mv).

If the mass of the mechanism is constant, then Δ(mv) = mΔv.

So the for consistent mass, Newton’s 2nd law of movement becomes

displaystylemathbfF_ extnet=fracDeltamathbfpDelta t=fracmDeltamathbfvDeltat

Because fracDeltamathbfvDeltat=mathbfa\, we gain the familiar equation Fnet = ma when the massive of the system is constant.

Newton’s 2nd law of motion declared in regards to momentum is more generally applicable because it have the right to be used to solution where the massive is changing, such together rockets, and also to equipment of consistent mass. Us will take into consideration systems v varying mass in part detail; however, the relationship between momentum and force remains beneficial when fixed is constant, such together in the complying with example.

Example 2. Calculating Force: Venus Williams’ Racquet

During the 2007 French Open, Venus Williams struggle the fastest taped serve in a premier women’s match, reaching a rate of 58 m/s (209 km/h). What is the average force exerted on the 0.057-kg tennis sphere by Venus Williams’ racquet, assuming the the ball’s speed just after influence is 58 m/s, the the early stage horizontal ingredient of the velocity before impact is negligible, and that the ball remained in call with the racquet because that 5.0 multiple sclerosis (milliseconds)?


This problem involves just one dimension since the ball starts from having actually no horizontal velocity component prior to impact. Newton’s 2nd law declared in terms of momentum is then written as

displaystylemathbfF_ extnet=fracDeltamathbfpDelta t

As provided above, as soon as mass is constant, the adjust in inert is offered by Δ p = mΔv = m ( v f − v i ) .

In this example, the velocity just after affect and the change in time space given; thus, as soon as Δp is calculated, mathbfF_ extnet=fracDeltapDelta t can be offered to discover the force.


To recognize the readjust in momentum, instead of the values for the initial and also final velocities right into the equation above.

eginarraylllDeltap&=&m(v_ extf-v exti)\ &=&(0.057 ext kg)(58 ext m/s-0 ext m/s)\ &=&3.306 ext kgcdot extm/sapprox3.3 ext kgcdot extm/sendarray\

Now the magnitude of the net exterior force can established by using mathbfF_ extnet=fracDeltapDelta t:

eginarraylllmathbfF_ extnet&=&fracDeltapDeltat=frac3.306 ext kgcdot extm/s5.0 imes10^-3 ext s\ &=&661 ext Napprox660 ext Nendarray\

where we have retained only two far-reaching figures in the final step.


This quantity was the average pressure exerted by Venus Williams’ racquet ~ above the tennis ball throughout its brief influence (note that the ball also experienced the 0.56-N pressure of gravity, yet that force was not because of the racquet). This difficulty could likewise be resolved by an initial finding the acceleration and also then utilizing Fnet = ma, however one extr step would be required compared with the strategy used in this example.

Section Summary

Linear momentum (momentum because that brevity) is identified as the product that a system’s mass multiplied by the velocity.In symbols, linear momentum p is characterized to be pmv, where m is the fixed of the system and v is the velocity.The SI unit for momentum is kg · m/s.Newton’s 2nd law of motion in terms of momentum states that the net exterior force equals the change in momentum of a system divided by the moment over which that changes.In symbols, Newton’s 2nd law of activity is defined to be mathbfF_ extnet=fracDelta mathbfpDelta t\Fnet is the net external force, Δp is the readjust in momentum, and Δt is the change time.

Conceptual Questions

An thing that has actually a tiny mass and an object that has a large mass have the very same momentum. Which object has the largest kinetic energy?An object that has a small mass and things that has a huge mass have the very same kinetic energy. I beg your pardon mass has actually the biggest momentum?Professional Application. Football coaches recommend players to block, hit, and also tackle with their feet top top the ground rather than through leaping through the air. Using the principles of momentum, work, and energy, explain how a football player have the right to be more effective through his feet ~ above the ground.How deserve to a small force impart the exact same momentum to things as a large force?

Problems & Exercises

(a) Calculate the inert of a 2000-kg elephant charging a hunter in ~ a speed of 7.50 m/s. (b) Compare the elephant’s momentum v the momentum of a 0.0400-kg tranquilizer dart fired in ~ a speed of 600 m/s. (c) What is the momentum of the 90.0-kg hunter running at 7.40 m/s after absent the elephant?(a) What is the fixed of a big ship that has actually a momentum of 1.60 × 109 kg · m/s, when the delivery is relocating at a speed of 48.0 km/h? (b) Compare the ship’s inert to the inert of a 1100-kg artillery covering fired in ~ a speed of 1200 m/s.(a) At what speed would a 2.00 × 104-kg airplane have to fly to have actually a inert of 1.60 × 109 kg · m/s (the exact same as the ship’s momentum in the trouble above)? (b) What is the plane’s momentum when it is acquisition off in ~ a speed of 60.0 m/s? (c) If the delivery is an plane carrier that launches this airplanes through a catapult, comment on the effects of her answer come (b) as it relates come recoil impacts of the catapult top top the ship.(a) What is the inert of a garbage truck that is 1.20 × 104 kg and also is relocating at 10.0 m/s? (b) At what speed would an 8.00-kg trash can have the exact same momentum as the truck?A runaway train automobile that has a massive of 15,000 kg travels at a speed of 5.4 m/s under a track. Compute the time forced for a force of 1500 N to carry the vehicle to rest.The mass of planet is 5.972 × 1024 kg and also its orbit radius is an median of 1.496 × 1011 m. Calculate its linear momentum.

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linear momentum: the product the mass and also velocity

second regulation of motion: physical regulation that claims that the net outside force equals the change in momentum of a system split by the time over which that changes

Selected solutions to Problems & Exercises

1. (a) 1.50 × 104 kg ⋅ m/s; (b) 625 to 1; (c) 6.66 × 102 kg ⋅ m/s

3. (a) 8.00 × 104 m/s; (b) 1.20 × 106 kg · m/s; (c) Because the momentum of the aircraft is 3 orders of magnitude smaller sized than of the ship, the ship will not recoil really much. The recoil would certainly be −0.0100 m/s, i beg your pardon is more than likely not noticeable.