Momentum is a measure of how much motion an object has. It depends on how much mass is moving and how fast it’s moving. The more of each, the more momentum, and the harder the object is to stop.
p = m v
The symbol p stands for momentum. Mass m is in kilograms and velocity v is in meters per second, so momentum is measured in kg·m/s. There is no special name for this unit, the way a joule has one for energy.
🔑Notice the word velocity, not speed. Velocity includes a direction. That makes momentum a vector, and that one fact is what makes collisions so interesting.
BConcept
Momentum Has a Direction
Kinetic energy doesn’t care which way an object moves, because the velocity is squared. Momentum does care. In a straight line, you handle direction with signs. Pick a direction to be positive, usually right or forward. Anything moving the opposite way gets a negative velocity, and therefore a negative momentum.
To find the total momentum of a group of objects, add their momenta with their signs:
p_total = p₁ + p₂ + p₃ + …
💡This is why two carts rolling toward each other can have a total momentum of zero. One is positive and one is negative, and the two cancel. It’s the first hint of why momentum is so useful for collisions, which you’ll explore in Activity 5-1-3.
CMath
Calculating Momentum
Example 1: find the momentum
Given
A car with mass m = 1,200 kg
Velocity v = +5.0 m/s
Solve
p = m v
p = (1,200 kg)(5.0 m/s)
p = 6,000 kg·m/s
✓ The car’s momentum is +6,000 kg·m/s.
Example 2: find the velocity
Given
A sprinter has mass m = 70 kg
Her momentum is p = 700 kg·m/s
Solve
p = m v so v = p ÷ m
v = 700 ÷ 70
v = 10 m/s
✓ She is running at 10 m/s.
Example 3: total momentum of two vehicles
Given
A 2,000 kg truck moves at +10 m/s
A 1,000 kg car moves toward it at −15 m/s
Solve
p_truck = 2,000 × (+10) = +20,000 kg·m/s
p_car = 1,000 × (−15) = −15,000 kg·m/s
p_total = +20,000 + (−15,000)
p_total = +5,000 kg·m/s
✓ The pair has a net momentum of +5,000 kg·m/s, in the truck’s direction.
DExample
Momentum Explorer
Move the sliders and watch the signed bars. Make the two carts’ momenta cancel, and then look at the kinetic energy.
Two carts, A and B. Positive velocity means moving right, and negative means moving left. Set their masses and velocities and watch the momentum bars. Try making the total zero.
The carts move in opposite directions, so their momenta partly cancel. Their kinetic energies still add.
EMath
Momentum vs. Kinetic Energy
Momentum and kinetic energy both depend on mass and speed, but not in the same way:
Change
Momentum becomes
Kinetic energy becomes
Mass ×2, same speed
×2
×2
Speed ×2, same mass
×2
×4
Speed ×3, same mass
×3
×9
Speed ×½, same mass
×½
×¼
Mass ×3, speed ×½
×1.5
×0.75
That difference is why the two can rank objects differently. Here are the baseball and the bowling ball again:
Object
Mass
Speed
Momentum
KE
Baseball pitch
0.145 kg
40 m/s
5.8 kg·m/s
116 J
Bowling ball, rolling
6.0 kg
6.0 m/s
36 kg·m/s
108 J
The two quantities are connected. Since p = mv, you can write v = p ÷ m and substitute it into the kinetic energy formula:
KE = ½ m v² = p² ÷ (2m)
Same momentum, different energy
Given
Object 1: m = 2.0 kg, p = 12 kg·m/s
Object 2: m = 6.0 kg, p = 12 kg·m/s
Solve
KE = p² ÷ (2m)
Object 1: KE = 12² ÷ (2 × 2.0) = 144 ÷ 4.0 = 36 J
Object 2: KE = 12² ÷ (2 × 6.0) = 144 ÷ 12 = 12 J
✓ With the same momentum, the lighter object has three times as much kinetic energy.
FWatch Out
Three Common Mix-Ups
⚠️“Momentum is just another word for kinetic energy.” They’re different quantities. Momentum has a direction and can be negative, and it’s proportional to speed. Kinetic energy has no direction, is never negative, and is proportional to speed squared.
⚠️“The heavier object always has more momentum.” Only if the speeds are similar. A 0.145 kg baseball at 40 m/s (5.8 kg·m/s) has more momentum than a 6.0 kg ball rolling at 0.5 m/s (3.0 kg·m/s). Always use both numbers.
⚠️“Momentum is a force.” It isn’t. A force is measured in newtons and momentum in kg·m/s. A moving object doesn’t push with its momentum. Rather, changing an object’s momentum takes a force acting for some time, which is the subject of Activity 5-1-2.
GConcept
Why a Big Momentum Is Hard to Stop
You already know from Unit 2 that a net force changes an object’s velocity: F = ma. Since a = Δv ÷ t, you can rewrite that as F = m × Δv ÷ t, and because m × Δv is the change in momentum:
F = Δp ÷ t
To bring an object to a stop, its momentum has to go to zero. The bigger its momentum, the bigger the force, the longer the time, or both. Here is what that looks like for a truck with 20,000 kg·m/s of momentum:
Preview: stopping the same truck two ways
Given
A truck’s momentum is 20,000 kg·m/s
It is brought to a stop in a long time or a short time
Solve
Stop over 4.0 s: F = 20,000 ÷ 4.0 = 5,000 N
Stop over 0.10 s: F = 20,000 ÷ 0.10 = 200,000 N
✓ The same change in momentum takes 40 times the force when the stop is 40 times quicker.
🔑Hold onto this idea. Stretching out the time of a stop shrinks the force. It’s the whole secret behind padded bumpers, airbags, and the egg-drop container you’ll design in Project 5-1-4.
HConcept
Quick Check
Four questions. They are not graded. Answer them to find out what still needs work.
Q1A 2.0 kg cart moves left at 3.0 m/s. Taking right as positive, what is its momentum?
Q2Cart A has mass 3.0 kg and velocity +4.0 m/s. Cart B has mass 2.0 kg and velocity −5.0 m/s. What is their total momentum?
Q3A cart’s speed doubles and its mass stays the same. What happens to its momentum and its kinetic energy?
Q4Two objects have the same momentum, but one has four times the mass of the other. Which has more kinetic energy?
Score: 0 of 4
IConcept
Looking Ahead
You can now count momentum and add it up with signs. Next you’ll look at what it takes to change it.