Projectile Motion in a Uniform Gravitational Field

A projectile moving under gravity alone moves horizontally and vertically at the same time. These components can be analysed independently: gravity changes the vertical motion but does not change the horizontal motion. Air resistance is neglected.

In a uniform gravitational field, the downward acceleration has a constant magnitude $$g$$. Taking upwards as positive, the accelerations are $$a_x=0$$ and $$a_y=-g$$. The horizontal velocity therefore remains constant, while the vertical velocity changes by the same amount each second.

Resolving the launch velocity

For a projectile launched at speed $$u$$ at an angle $$\theta$$ above the horizontal, the initial velocity components are:

The horizontal component remains $$u\cos\theta$$ throughout the flight. The vertical component starts at $$u\sin\theta$$ and decreases as gravity acts.

Position and velocity during flight

Measure horizontal and vertical displacement from the launch point, with upwards positive. After a time $$t$$, the displacements are:

The horizontal displacement increases uniformly because horizontal velocity is constant. Vertically, the term $$u\sin\theta,t$$ represents the displacement from the initial vertical velocity, while $$-\frac{1}{2}gt^2$$ accounts for the downward acceleration.

At the same time $$t$$, the velocity components are:

The uniform-acceleration equations apply separately to each component. The motions are independent, but they share the same elapsed time: a value of $$t$$ gives both components of position and both components of velocity at that instant.

Highest point and trajectory

As the projectile rises, its vertical velocity decreases. At the highest point, $$v_y=0$$ for an instant, but the horizontal velocity need not be zero. The downward acceleration is still $$g$$, so the vertical velocity then becomes negative as the projectile descends.

When the horizontal velocity is non-zero, the combination of constant horizontal velocity and constant downward acceleration produces a parabolic trajectory. If the projectile is launched vertically, its path is instead a straight vertical line.

Friction

Friction is a contact force between surfaces that touch. It acts parallel to the surfaces and opposes their relative motion, or their tendency to move relative to one another.

When a box slides across the floor, friction on the box acts opposite to its sliding motion. This can slow the box down. If the box is pushed but does not move, friction can balance the push and prevent the box from slipping.

The magnitude of friction depends on the nature of the surfaces and on how strongly they are pressed together. For surfaces that are not slipping, the frictional force also depends on the forces trying to make them move relative to each other.

Friction does not always stop an object moving. When a person walks, their foot pushes backwards against the ground. Friction prevents the foot from slipping backwards and pushes the person forwards.

Lift

A fluid, such as a gas or liquid, can exert a force on an object moving relative to it. Lift is the component of that force acting perpendicular to the relative flow: the direction in which the fluid moves past the object.

Lift can arise from pressure differences around the object. The fluid pushes on different parts of the surface, and these pushes combine to produce a resultant force. If that resultant has a component perpendicular to the relative flow, that component is lift.

The flow around an object depends on the object’s shape and orientation, as well as on the flow itself. Changing these can change the pressure distribution and therefore the lift.

Lift is defined by the flow direction, not by the vertical. For example, if the relative flow is vertical, the lift acts horizontally. It is therefore not necessarily an upward force.

Drag and Air Resistance

An object moving through a fluid experiences a force that opposes its motion relative to that fluid. This force is called drag. A fluid can be a liquid or a gas; air resistance is the drag exerted by air.

Drag depends on the motion of the object relative to the fluid, not just on its motion relative to the ground. For example, an object held still in a wind experiences air resistance because air is moving past it.

What affects air resistance?

Air resistance increases as the object's speed relative to the air increases. It also depends on the object's shape and the area presented to the airflow. For similar shapes moving at the same speed relative to the air, a larger area facing the airflow generally produces greater air resistance.

Direction of drag

Drag acts opposite to the object's motion relative to the fluid. It can therefore change the object's velocity: its speed, its direction, or both.

In still air, air resistance acts opposite to the object's motion and reduces its speed. If the air is moving sideways relative to the ground, the drag force can also act sideways relative to the object's path, changing its direction.

Air Resistance and Projectile Trajectories

A projectile moving through air experiences air resistance in the direction opposite to its velocity relative to the air. In still air, this means the force opposes the projectile’s motion at each point in its flight. Gravity continues to act vertically downwards.

How air resistance changes the motion

For a projectile travelling horizontally through still air, air resistance has a horizontal component that opposes its motion throughout the flight. Its horizontal velocity therefore decreases rather than remaining constant.

The vertical effect changes as the projectile rises and falls:

• During ascent, air resistance acts partly downwards, opposing the upward motion. Together with gravity, it slows the upward motion more quickly than gravity alone would.
• At the highest point, the vertical velocity is momentarily zero. If the projectile is still moving horizontally, air resistance still acts horizontally.
• During descent, air resistance acts partly upwards, opposing the downward motion. It counteracts some of gravity’s downward effect.

As the projectile’s speed and direction change, so does the effect of air resistance. The vertical acceleration is therefore not a constant downward $$g$$ throughout the flight, even though the acceleration due to gravity itself remains approximately constant.

Trajectory compared with gravity alone

Compared with a projectile launched in the same way and moving under gravity alone, a projectile affected by air resistance reaches a lower maximum height and travels a shorter horizontal distance, when landing at the same level. Its path is no longer a parabola or symmetric about its highest point.

Because the acceleration changes during flight, the uniform-acceleration equations cannot describe the complete motion using one constant acceleration.

Terminal Speed

A moving object reaches terminal speed when the resistive force opposing its motion increases enough to balance the driving force. The resultant force is then zero, so the object stops accelerating and continues at a constant speed.

Terminal speed of a falling object

A falling object has two opposing forces: its weight acts downwards, while air resistance acts upwards. When the object first starts to fall, air resistance is small, so weight is greater. The resultant force is downwards and the object accelerates downwards.

As the object gains speed, air resistance increases. This reduces the resultant downward force, so the object continues to speed up but accelerates less rapidly.

Eventually, air resistance becomes equal to weight. The forces are balanced, so the resultant force and acceleration are both zero. The object does not stop: it continues falling at its constant terminal speed.

Maximum Speed of a Vehicle

A vehicle accelerates when its driving force is greater than the total resistive force acting against its motion. As its speed increases, air resistance increases, leaving a smaller resultant forward force.

The vehicle reaches its maximum steady speed when the driving force available at that speed balances the total resistive force. The resultant force is then zero, so the vehicle no longer accelerates and continues at a steady speed.

What determines the maximum steady speed?

The maximum steady speed depends on the driving force available from the vehicle’s propulsion and on the resistive forces, including air resistance and friction. More driving force is needed to continue accelerating as air resistance increases.

Air resistance depends partly on the vehicle’s shape and the area it presents to the airflow. A streamlined shape or a smaller frontal area can reduce air resistance. If a vehicle has greater resistance at a given speed, it reaches a lower maximum steady speed for the same available driving force, with other conditions unchanged.