# Lesson Explainer: Newton’s First Law of Motion Mathematics

In this explainer, we will learn how to solve problems using Newton’s first law.

Let us begin by defining Newton’s first law of motion.

### Definition: Newton’s First Law of Motion

A body in uniform motion will continue in that uniform motion unless a nonzero net force acts on it.

For the motion of a body to be uniform, the velocity of the body must be constant. A body that is at rest is a special case of a body in uniform motion, where the speed of the body is constantly zero.

If multiple forces act on a body, the net force due to these forces may be zero. A familiar example of this is the case of a body at rest on a smooth, horizontal surface. The forces acting on such a body are shown in the following figure.

The force is the weight of the body, and the force is the normal reaction force due to the contact of the body and the surface. For a body at rest, the magnitudes of and must be equal, and and must act in opposite directions with the same line of action.

The phenomenon of a body remaining at rest or at a uniform velocity when no net force acts on it, or Newton’s first law of motion, is also known as inertia.

### Definition: Inertia

Inertia is a property of all matter by which an object maintains its current state of rest or uniform linear motion unless acted upon by an external force.

There is a relationship between the amount of inertia a body has and that body’s mass.

Imagine you had balls lying in front of you. One is a Ping-Pong ball with a mass of 3 grams and the other is a bowling ball with a mass of 8 kilograms. If you were required to roll the two balls away from you, it would be considerably harder to roll the bowling ball than the Ping-Pong ball. In other words, the force required to start moving the bowling ball is a larger than that required to start moving the Ping-Pong ball.

This happens because the bowling ball endeavors to maintain its current state of rest more than the Ping-Pong ball. In other words, it has a greater inertia, which is due to the fact that it has a greater mass.

### Property: The Relationship between Inertia and Mass

The greater a body’s mass, the greater the force required to move it from rest. In other words, the greater the mass of the body, the more inertia the body has.

Let us look at an example of a body in uniform motion that is acted on by multiple forces.

### Example 1: Finding Missing Forces Acting on a Body Moving at a Constant Speed Using Newton’s First Law

In the figure, the body is moving at a constant velocity under the action of a system of forces. Given that the forces are measured in newtons, find the magnitudes of and .

The body moves at constant velocity, and so the net force acting on it must be zero.

The net forces acting on the body parallel to and perpendicular to can be considered separately.

Perpendicularly to , a force of 20 N and a force of 31 N act in the same direction. The magnitude of must be the sum of these forces, so

Parallel to , the forces acting on the body are the 79 N force and . The magnitude of must, therefore, be 79 N.

Let us look at another such example.

### Example 2: Finding Missing Forces Acting on a Body Moving at a Constant Speed Using Newton’s First Law

In the given figure, the body is subject to the action of a system of forces. Given that it is moving at a constant speed and that the forces are measured in newtons, find and .

The body moves at constant velocity, and so the net force acting on it must be zero.

The net forces acting on the body parallel to and perpendicular to can be considered separately.

Perpendicularly to , a force of 35 N and a force of 31 N act in opposite directions. The magnitude of must be the difference between these forces, so

Parallel to , the forces acting on the body are the 56 N force and . The magnitude of is given by

Let us now look at an example where multiple forces act on a uniformly moving body and one of the forces acts in a direction neither parallel nor perpendicular to the velocity of the body.

### Example 3: Finding Missing Forces Acting on a Body Moving at a Constant Speed Using Newton’s First Law

A body of mass 20 kg is pulled along a horizontal plane by a rope that makes an angle with the plane, where . When the tension in the rope is 91 N, the body moves with uniform velocity. Find the total resistance to the motion, , and the normal reaction, . Use .

The forces acting on the body are shown in the following figure.

The body has a constant velocity, so the net force on the body is zero.

For the net force on the body parallel to the velocity of the body to be zero, it must be the case that

For the net force on the body perpendicular to the velocity of the body to be zero, it must be the case that

The rope that supplies the 91 N force acts at an angle from the horizontal. As is stated,

The rope corresponds to the hypotenuse of a right triangle with the legs shown in the following figure.

The length of the hypotenuse, , is given by

From this, we can determine that and

These values allow and to be determined as follows:

Newton’s first law of motion can be applied in a context where a varying force must take a particular value required to produce uniform motion. Let us look at such an example.

### Example 4: Finding the Maximum Speed in a Real-World Problem Using Newton’s First Law

A soldier jumped out of a plane with a parachute. After he had opened his parachute, the resistance to his movement was directly proportional to the cube of his speed. When his speed was 19 km/h, the resistance to his motion was of the combined weight of him and his parachute. Determine the maximum speed of his descent.

The falling soldier is accelerated downward by the gravitational force acting on him, increasing his downward velocity. The upward-acting resistance to the motion of the soldier varies with the velocity of the soldier.

As the soldier falls and accelerates downward, the resistance to his motion increases, decreasing the net downward force on him, so also decreasing his downward acceleration, hence decreasing the rate at which his downward velocity increases.

The maximum speed of the soldier’s descent is reached at the instant his downward velocity becomes constant. At that instant, the acceleration of the soldier must be zero, and so according to Newton’s first law of motion, the net force on the soldier must be zero at that instant.

It is stated that when the speed of the soldier is 19 km/h, the resistance force produced has a magnitude of where is the weight of the soldier plus parachute.

It is stated that the resistance acting on the soldier plus parachute is directly proportional to the cube of the speed of the soldier, which gives us where is a constant of proportionality.

At the instant that the magnitude of the resistance acting on the soldier plus parachute is equal to the magnitude of the weight of the soldier plus parachute, the resistance force magnitude, , is equal to , so it is 27 times the magnitude of the resistance force at 19 km/h. When the resistance increases by a factor of 27, also increases by a factor of 27 to give the maximum velocity, . This gives us

The value of is found by dividing both sides of the equation by and taking the cube root of the result:

### Key Points

• A body in uniform motion will continue in that uniform motion unless a nonzero net force acts on it.
• A body at rest is a special case of uniform motion for which the speed of the body is constantly zero.
• If multiple forces act on a body in uniform motion, the resultant of these forces must be zero.