Motion with non-uniform acceleration (3.2.2)

When an object moves through a fluid (liquid or gas), such as the air or water, it experiences a drag force due to the fluid.

Drag is a frictional force that opposes the motion of an object moving through the fluid, slowing the object down. Therefore, drag forces always act in the opposite direction to the object’s motion.

The drag force converts some of the object’s kinetic energy into thermal energy within the fluid due to the work done against the resistance of the fluid.

For example, some objects entering the Earth’s atmosphere from space burn up due to the very large speeds at which they travel. The object experiences a very large drag force, which results in large amounts of thermal energy.

The magnitude of the drag force is dependent on multiple factors. These include:

• the speed of the object
• the shape of the object
• the surface characteristics of the object, e.g. roughness
• the density of the fluid
• the cross-sectional area of the object.

The following equation of drag force is not required knowledge for your A-level study. However, the formula for drag, for a smooth object in a fluid of uniform density is given below for your understanding:

Where:

• is the drag coefficient (dimensionless, depends on the shape of the object)
• is the density of the fluid
• is the cross-sectional area of the object
• is the object’s speed.

The two factors that have the most significant impact on the drag force are the speed and cross-sectional area of the object.

The cross-sectional area of an object is the area of the shape formed when the object is cut perpendicular to a specific axis.

In the cylinder example above, the cross-sectional area would be the area of a circle, i.e.

When discussing drag force, the relevant cross-sectional area is the area of the shape formed perpendicular to the direction of motion. This represents the ‘face’ presented to the fluid, which will create the drag force on the object.

The two most important factors affecting the magnitude of the drag force are the object’s speed and the cross-sectional area: a larger cross-sectional area results in a greater drag force.

For most objects moving through a fluid, the drag force is directly proportional to the speed of the object squared, i.e.

As the graph shows above, if we double the speed, the drag force quadruples.

When objects move through the air, they experience a drag force known as air resistance.

For example, cars and aeroplanes typically have smooth and streamlined shapes to reduce the amount of air resistance they experience. This allows the vehicle to travel at higher speeds while also reducing the amount of fuel consumed.

Air resistance significantly affects the motion of a projectile through the air.

The sketch graph shows the difference in the height and range of a projectile without and with the presence of air resistance.

The factors affected by air resistance are as follows.

• Height: without air resistance, a projectile follows a symmetric parabolic path. However, with air resistance, the vertical velocity component decreases at a greater rate as the projectile rises, reducing the maximum height it can reach.

• Range: air resistance slows the projectile throughout its flight, reducing its horizontal velocity. As a result, the projectile covers less horizontal distance before landing, shortening the range compared to motion in a vacuum.

When an object is in free fall through a fluid, its weight remains constant throughout the fall. However, the drag force increases as the object’s speed increases.

At the instant an object begins to fall, the drag force is zero, and the total force acting on the object is due to its weight. The object accelerates at a rate equal to the acceleration due to free fall.

The image above shows that as the object falls, the speed increases, and so does the magnitude of the opposing drag force. The resultant force (net force) on the object decreases, and the instantaneous acceleration of the object decreases to less than

Eventually, the object will achieve constant speed due to the force of the weight and the force of the drag becoming equal. This is known as terminal velocity.

When an object is in free fall, its weight and drag force eventually become equal, and the object falls at a constant velocity known as terminal velocity.

The sketch graph below shows a velocity–time graph for an object in free fall through the air.

The object’s weight is equal to and the drag force is equal to The instantaneous acceleration of the object is The resultant force changes at each of the three times:

• At the only force acting on the object is the weight, so the resultant force is equal to Therefore,

• At the resultant force equals the difference between the weight and the drag force. Therefore:

• At terminal velocity has been reached, i.e. Therefore, the resultant force is zero and

To investigate the motion of an object falling through a fluid under the influence of a drag force, you can use a motion sensor connected to a data logger or laptop.

In the setup below, a thin string passed over a pulley attaches the falling object to a light polystyrene ball. As the object falls through a liquid cylinder, such as water or glycerol, it pulls the polystyrene ball upwards.

The motion of the polystyrene ball in the air is identical to that of the object falling through the fluid, allowing you to analyse velocity–time and acceleration–time graphs without directly measuring the object’s motion in the liquid.

Pointing the motion sensor directly at the falling object in the fluid is not practical due to several limitations.

• The liquid can distort or scatter the sensor’s signal, resulting in inaccurate measurements.
• Small objects may be difficult for the sensor to track reliably, especially in viscous fluids where turbulence or bubbles can interfere.

Monitoring the polystyrene ball in the air eliminates these issues, as the sensor operates more effectively in the air and provides clean, accurate data while still reflecting the object’s motion in the fluid.

The graph below shows the velocity of a skydiver who deploys a parachute over time.

1. The skydiver initially accelerates during free fall and eventually reaches terminal velocity as the force of air resistance balances the skydiver’s weight.
2. When the parachute is deployed, the skydiver immediately begins to decelerate, and their velocity decreases due to the much greater surface area and a large increase in air resistance.
3. With the parachute deployed, the skydiver continues decelerating until they reach terminal velocity again. The increased air resistance caused by the parachute enables the skydiver to reach a significantly lower terminal velocity, allowing them to land safely.