Push a small cart, then take your hand away. The cart keeps moving for a while. What is pushing it forwards now? “Some of the pushing force remains inside it” may sound natural, but it gets in the way of explaining motion. The push has ended; the motion it changed continues.
Force describes an interaction between bodies, and the net external force changes the velocity of the body we choose to study. In classical mechanics, in an inertial reference frame, zero net force preserves the existing velocity: a body stays at rest or moves at a constant speed in a straight line. To understand the cart, we need to distinguish how it is already moving from what changes that motion. OpenStax: force, Newton’s first law.
For a force, name who acts and what they act on
While your hand touches the cart, it pushes on it. Earth pulls the cart downwards, and the table or rail supports it from below. These are different interactions. Force describes such an action quantitatively: both its magnitude and its direction matter. A push to the right and an equally strong push to the left have different effects.
Force is measured in newtons, abbreviated N. An arrow is useful in a diagram: its direction shows the direction of the force, and its length, on a chosen scale, shows the magnitude. A force arrow is neither a path nor a velocity arrow. Its direction alone cannot tell us which way the cart is moving now.
Velocity describes how fast and in which direction position changes. Acceleration describes the change in velocity over time. A body accelerates when it speeds up, slows down or turns. Force is related to that change in velocity. NASA Glenn: laws of motion.
First collect all the forces on one cart
Imagine a horizontal rail and a cart that does not bounce. Gravity pulls it downwards; the support pushes it upwards. In our model, these vertical forces balance. Now consider the horizontal forces. If one device pulls the cart to the right with a force of 2 N and another pulls it to the left with 2 N, their combined horizontal effect is zero.
This sum, including directions, is called the net force, or resultant force. If 3 N acts to the right and 1 N to the left, the net force is 2 N to the right. It summarizes forces already counted; we must not add it as one more separate force. Forces at an angle also require us to account for the geometry of their directions. OpenStax: adding external forces.
Zero net force does not mean there are no interactions. Two equal, opposite forces may act simultaneously. We also need to be clear which body we are adding forces on: here, only the cart. The forces it exerts on a hand or rail belong to the description of those other bodies.
With zero net force, the existing motion is preserved
Under equal, opposite forces, a stationary cart stays at rest. But a cart already moving to the right at 2 m/s keeps moving in the same way: zero net force does not erase its velocity. This is Newton’s first law, the law of inertia. OpenStax: inertia and the role of friction.
Why, then, does an everyday cart stop? After the push, the force from your hand disappears, but air resistance, losses in the wheels and other interactions can produce a net force against the motion. That force slows the cart. The smaller the total resistance, the longer its velocity remains almost unchanged. Motion does not consume a supply of “force inside”.
There is a condition concerning the observer. We describe motion relative to the table and, for this short experiment, treat that reference frame as approximately inertial: suitable for applying the first law. In a train that brakes sharply or turns, the description must account for the acceleration of the reference frame itself. We cannot silently carry our simple diagram over to that situation.
The same net force changes different masses differently
Now remove the pull to the left. A force of 2 N to the right remains. According to Newton’s second law, the cart accelerates in the direction of the net force. For the same mass, a larger net force produces a larger acceleration; for the same net force, a larger mass produces a smaller acceleration. Mass here characterizes a body’s inertia: how much a given force changes its motion. OpenStax: force, mass and acceleration.
Compare the five independent starting states below. In each, the forces act for one second, mass stays constant, and we neglect friction and air resistance. External devices provide the pull to the left or right. These are calculated consequences of a model, not measurements of real carts.
Five starting states — one second later
These are independent cases, not successive frames. All horizontal forces in each case remain constant for 1 s. Right is the positive direction.
1Balanced forces, initially at rest
Horizontal forces on the cart
Left: 2 N · Right: 2 N
- Mass
- 1 kg
- Initial velocity
- 0 m/s
- Net force
- 0 N
- Acceleration
- 0 m/s²
- Velocity after 1 s
- 0 m/s
The cart stays at rest.
2Balanced forces, initially moving
Horizontal forces on the cart
Left: 2 N · Right: 2 N
- Mass
- 1 kg
- Initial velocity
- 2 m/s →
- Net force
- 0 N
- Acceleration
- 0 m/s²
- Velocity after 1 s
- 2 m/s →
The motion is unchanged.
3Net force forwards
Horizontal forces on the cart
Left: 0 N · Right: 2 N
- Mass
- 1 kg
- Initial velocity
- 2 m/s →
- Net force
- 2 N →
- Acceleration
- 2 m/s² →
- Velocity after 1 s
- 4 m/s →
Speed increases by 2 m/s.
4The same net force, twice the mass
Horizontal forces on the cart
Left: 0 N · Right: 2 N
- Mass
- 2 kg
- Initial velocity
- 2 m/s →
- Net force
- 2 N →
- Acceleration
- 1 m/s² →
- Velocity after 1 s
- 3 m/s →
Speed increases by only 1 m/s.
5Net force against the motion
Horizontal forces on the cart
Left: 1 N · Right: 0 N
- Mass
- 1 kg
- Initial velocity
- 2 m/s →
- Net force
- 1 N ←
- Acceleration
- 1 m/s² ←
- Velocity after 1 s
- 1 m/s →
The cart slows down but still moves right.
The first two cases have the same zero net force but different velocities. The third and fourth show the role of mass: 2 N changes the velocity of a 1 kg cart by 2 m/s in a second, but that of a 2 kg cart by only 1 m/s in the same second. This is about a change in velocity, not a rule that “heavier always moves more slowly”.
In the fifth case, force acts to the left even though the cart is moving to the right. In one second, its speed falls from 2 to 1 m/s. It is not moving backwards yet. If the same force continues, the cart first reaches zero velocity and only then starts moving left. Acceleration and existing motion can point in different directions.
Branch · With formulasOptional · you can skip this section
Calculating what happens in one second
For the translational motion of a constant-mass body in our inertial model:
\vec F_{\mathrm{net}}=m\vec a.
Here \vec F_{\mathrm{net}} is the net external force, m is mass and \vec a is acceleration. Arrows above symbols remind us that direction matters. One newton is the force that gives a 1 kg body an acceleration of 1 m/s². Thus 1 N = 1 kg ⋅ m/(s)^(2).
Along our straight line, choose right as positive. In the fifth case, F_(net) = − 1 N and m = 1 kg, so a = − 1 m/s². For constant acceleration over a time Δt:
v_(1) = v_(0) + aΔt; = 2 m/s; + ( − 1 m/(s)^(2)) ⋅ 1 s; = 1 m/s.
The minus sign in acceleration means leftwards; the positive velocity result means the cart is still moving right. With zero net force, a = 0, so v_(1) = v_(0) for any initial velocity. The equation does not choose the initial state for us.
Why an equal, opposite force does not cancel the push
When your hand pushes the cart, the cart simultaneously pushes your hand with an equal force in the opposite direction. This is Newton’s third law. How can the cart accelerate at all?
Because the two forces in this interaction pair act on different bodies. The hand’s force on the cart belongs in the cart’s net force. The cart’s force on the hand belongs in the hand’s net force. Adding them and declaring “everything cancels” would mean changing the chosen body halfway through the explanation. OpenStax: third-law forces.
Compare this with the vertical forces. Earth’s gravity and the rail’s support act on the same cart and balance in our situation. But they arise from different interactions and are not a third-law pair. The partner to Earth’s force on the cart is the cart’s force on Earth; the partner to the rail’s force on the cart is the cart’s force on the rail. Equal arrow lengths alone do not tell us how two forces are related.
The same principle helps explain a rocket: it expels gases backwards, and the gases exert a force forwards on it. This interaction does not require surrounding air for the rocket to push against. Calculating a rocket’s motion is more involved than our cart example, because its mass changes as propellant flows out. OpenStax: rocket thrust.
Branch · DeeperOptional · you can skip this section
Force can turn motion as well as speed it up
Our rail allowed motion only along a straight line. In space, a force can also act sideways relative to the current velocity. It then changes the direction of motion. Even if speed stays constant, this turning is acceleration.
In a model of uniform circular motion, the net force always points towards the centre, while velocity points along the tangent. These are different arrows with different meanings. Orbital motion shows why judging force only by whether a body is speeding up is insufficient. Real orbits can be noncircular; both direction and speed then change. OpenStax: force directed towards the centre.
Back to the cart after the push
A cart is moving right at 2 m/s. Two equal, opposite horizontal forces are applied to it. Will it stop just because the forces “balance”? In our model, no. The net force is zero, so its velocity stays at 2 m/s to the right. Slowing it down requires a net force against its motion.
The three laws form one way of reasoning: choose the body and reference frame, establish its initial motion, find all external forces on it, and use the net force to determine the change in velocity. Look for the equal, opposite third-law force on the interaction partner.
We have applied classical mechanics to slow-moving macroscopic bodies with constant mass. We have not described the cart’s rotation or deformation, quantum phenomena, or motion at speeds close to the speed of light. Those questions require additional concepts and different model limits. OpenStax: the domain of Newtonian mechanics.
After the push, the cart moves because it already has velocity. Further interactions change that velocity; an internal “force of motion” does not run out. The course’s next question concerns energy: what changes, and where does it go, when the cart speeds up, brakes or heats a surface?