Terminal velocity

In a nutshell

Terminal velocity is the maximum possible velocity an object has as it travels through a fluid. When an object has reached terminal velocity the resultant forces acting on the object is zero.

Equations

DESCRIPTION	SYMBOL EQUATION
Drag	$F = ma$
Drag for a falling object	$W=mg$

Constants

CONSTANT	SYMBOL	VALUE
$acceleration\ due \ to \ gravity$	$g$	$9.81 \, ms^{-2}$

Variable Definitions

QUANTITY NAME	SYMBOL	DERIVED UNIT	SI BASE UNITS
$force$	$F$	$N$	$kgms^{-2}$
$mass$	$m$	$kg$	$kg$
$weight$	$W$	$N$	$kgms^{-2}$
$acceleration$	$a$	$m\,s^{-2}$	$m\,s^{-2}$

Drag

An object moving through a fluid feels a resistive force acting on it called drag. Drag is a frictional force opposing motion. The force depends on multiple factors such as: the speed of the object, the cross-sectional area of the object and the density of the fluid.

When an object is falling through a uniform gravitational field, it is assumed to have a uniform attractive force acting on it:

$W= mg$

Drag forces oppose the force of gravity, and therefore decrease the net force acting on an object. The drag force will increase until it equals the force from the weight of the object, resulting in terminal velocity.

If an object is not under freefall then the generalised equation of Newtons second law can be used:

$F=ma$

Curiosity: fast cars are very low profile and streamlined to reduce the air resistance acting on them.

Terminal velocity

As the velocity increases due to free fall, the drag force also increases. As the weight remains constant, the net force will decrease until it reaches zero. Throughout this period, acceleration will also decrease, as this is proportional to the force.

At the point where the net force is zero, the object will no longer accelerate, and is said to have reached its terminal velocity. It will remain at this velocity until it reaches the ground.

Terminal velocity of a skydiver

Terminal velocity is the maximum possible velocity an object has as it travels through a fluid. An example of terminal velocity is that of a skydiver jumping out of a plane:

1	As soon as the skydiver jumps out of the plane, the skydiver begins to accelerate as their force due to gravity (weight) is much larger than the air resistance.
2	The skydiver's downward velocity rapidly increases, which increases the air resistance. This causes the downwards acceleration of the skydiver to slowly decrease.
3	The downwards acceleration of the skydiver decreases gradually, until the point where the driving force (weight) is equal to the air resistance. The skydiver has reached terminal velocity.
4	The skydiver opens their parachute. This causes the air resistance acting on the skydiver to sharply increase, as the cross-sectional area of the object has increased. The weight of the skydiver has not changed. The force due to gravity is now much less than the air resistance, so the skydiver begins to decelerate.
5	As the skydiver's velocity decreases, so does the air resistance acting on them. The air resistance decreases until it is equal to the skydiver's weight. The skydiver has reached a new, lower terminal velocity.
6	The skydiver falls slowly to the ground.

In summary, skydivers use parachutes to greatly increase the drag acting on them, which acts to decrease their terminal velocity to a value which is safe enough to fall to the ground to.

Moving vehicles can also have terminal velocity as they travel through the air. For a car, the driving force is generated by the engine, and the resistive forces are friction and air resistance.

Note: The amount of drag that is generated depends on the cross-sectional area of the object.

Example

Below is a graph of a shuttlecock falling after it reached $0 \ ms^{-1}$ after it had been hit. Initially, the resultant force was equal to that of it's weight. Then, the drag force increased until both drag and weight were equal. The shuttlecock then reached terminal velocity.

Physics; Fluids; KS5 Year 12; Terminal velocity

Determining terminal velocity in a fluid

The terminal velocity of a viscous fluid can be determined experimentally. A viscous fluid is placed in a cylinder, marks are made on the cylinder at equal intervals.

A ball bearing is released from rest into the liquid and the timer is started. When the ball bearing passes a mark, the stopwatch is lapped.

The distances between the marks is known and as the time is collected from the stopwatch, so the velocity can be calculated.

When the ball bearing has reached terminal velocity, the velocity of the ball will become constant.

Physics; Fluids; KS5 Year 12; Terminal velocity

1.	Ball bearing enters the viscous fluid.
2.	The distance between markers are at regular intervals.
3.	The cylinder.
4.	Ball reaches terminal velocity.
5.	The bottom of the cylinder.