Newton's laws of motion (3.5.1)
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Newton’s First Law of Motion is the law of inertia. It states that an object at rest or moving with constant velocity will remain so unless acted upon by a net external force.
The key idea is that objects resist change unless a net external force acts upon the object. Resisting changes in motion is called inertia.
An object moving at constant velocity has no overall resultant force.
In the diagrams above, the overall net force is zero. Let the pull force be and friction be
Note the angular dependence on the pull force. Acting at an angle, only the horizontal pull force component acts in the same direction as velocity and against friction.
The vertical component of the pull force acts in the same direction as the normal force from the surface. These combined forces act equally and opposite to the force due to gravity, resulting in a vertical force of zero.
Newton’s Second Law of Motion states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass.
where:
- is the force in ,
- is mass in , and
- is acceleration in .
The key idea is that greater forces acting on an object produce a greater acceleration. Conversely, a more massive object requires a greater force to achieve the same acceleration as an object with less mass.
Newton’s Third Law of Motion is the law of action and reaction. It states that any action produces an equal and opposite reaction.
The key idea is that forces occur in pairs: the action of exerting a force produces an equal and opposite force. Equal and opposite are used to describe forces that have the same magnitude but are acting in opposite directions:
If only considering the magnitude of forces in a pair, and not direction, then:
Momentum describes how much motion an object has and its resistance to change in velocity:
Where:
- is the momentum in ,
- is mass in , and
- is velocity in .
Momentum is proportional to velocity. Velocity is a vector with a magnitude component (speed) and a direction. Hence, momentum is also a vector with both magnitude and direction.
Scalar quantities only have magnitude, such as mass and speed, and therefore do not change with direction.
The laws of classical mechanics dictate that momentum is always conserved. This means that the total momentum of all interacting bodies before and after a collision is the same.
The conservation of momentum can be used to calculate the velocity of objects before and after a collision. For instance, two objects colliding as illustrated below:
All forces are vectors. Finding a resultant force can be done by determining the horizontal and vertical components of all forces acting on an object.
Forces that act only in the vertical have no horizontal component, and the same applies to forces that only act along the horizontal; there is no vertical component.
In the example above, a buoy on a lake is being pulled through the water. The forces with vertical components are weight, buoyancy and the pull force. Overall vertical forces:
This means that forces are balanced in the vertical direction, with no overall vertical force.
Forces acting horizontally are the drag and pull.
Therefore, the overall force in the horizontal direction is .
The net force on an object is defined as the change in momentum over a period of time:
Where:
- is the net force in ,
- is the change in momentum in and
- is the change in time in .
Newton’s Second Law is a specific case of net force where the object’s mass remains constant. However, what if the mass of an object with a net force applied to it changes with time? Then, only the general form is applicable:
Rockets ejecting fuel as they accelerate are a good example of this. The rocket loses mass as it accelerates.
At time the rocket has mass , but after a period of time , fuel of mass has been ejected from the rocket, making it less massive. The net force on the rocket has remained constant during this time:
Since the change in momentum remains constant over time, the rocket’s velocity must increase with time as its mass decreases.
Newton’s second law is a specific case of net force where the object’s mass remains constant. It is possible to derive this equation from the general form where net force is the time derivative of momentum .
Substituting the equation for momentum removes from the equation.
Mass is not dependent on time, so it can be taken out of the front of the derivative. Velocity remains in the derivative as it is time-dependent.
The time derivative of velocity is acceleration:
By substitution, the final equation is achieved. Hence, Newton’s 2nd law is a specific case of net force .
Question walkthrough
Finding Mass Using Force and Momentum Change
Calculates a car's mass from driving force and change in velocity over time using F = dp/dt, without first finding acceleration.
Impulse is the effect of a force acting on an object over a given time period :
Where:
- is the impulse in ,
- is the force in and
- is the change in time in .
Impulse can be considered the change in momentum of an object.
A change in momentum is a result of a change in velocity at constant mass:
Where:
- is the impulse in ,
- is change in momentum in ,
- is mass in
- and is the change in velocity in .
An excellent illustration of impulse is rain versus hail when standing under an umbrella. Raindrops are smaller and have less mass, and when they hit an umbrella, they have a relatively small change in momentum.
On the other hand, hail has a larger mass, and when it hits an umbrella, there is a greater change in momentum and, therefore, impulse. Standing in hail with an umbrella, a person will feel significantly more force per second than in rain.
Question walkthrough
Impulse Delivered to a Returning Tennis Ball
Calculate the impulse and direction of the force delivered by a racket to a tennis ball that rebounds at a lower speed.
Impulse can be determined graphically. In a force-time graph, the area under the graph equals the impulse. This is particularly important in instances where force is not constant but varies over time.
Calculating the area under the graph can be achieved through geometry or integration.
Geometry: If, for example, a force-time graph is linear, then the area under the graph is a triangle and can be calculated from:
where:
- is the area in ,
- is the base in , and
- is the height in .
Integration: For more complicated graphs, integration can be used to find the area. For this, the equation of the force-time relationship is needed. To calculate the area, the limits of the integral are needed – these are the times and where the graph crosses the x-axis. Then, construct the integral:
which gives the area under the curve.
It is important to note that you you do not require the explicit use of derivatives or integrals to solve problems in your A-level exams. The above has been provided for your holistic understanding.
Force-time graphs can be used in everyday physical scenarios. For instance, they can highlight the importance of seat belts.
The peak force is significantly larger without the seatbelt, which is more hazardous. The seat belt increases the time over which the force is spread, reducing the peak. However, both graphs have the same area under them, so the impulse is the same.