Projectile motion and resistive forcesAQA A-Level Physics: Revision notes

In a uniform gravitational field, the horizontal and vertical motions of a projectile are independent:

• Horizontal: no force (ignoring air resistance), so the velocity is constant: $x = u_x t$.
• Vertical: constant acceleration $g = 9.81$ m s⁻² downwards, so the equations of uniform acceleration apply.

The time of flight is the same for both motions. An object thrown horizontally takes exactly as long to fall as one dropped from the same height.

1. Resolve the launch velocity into $u_x = u\cos\theta$ and $u_y = u\sin\theta$.
2. Use the vertical motion ($s = ut + \tfrac{1}{2}at^2$, $v^2 = u^2 + 2as$, with $a = -g$ if upward is positive) to find the time.
3. Use $x = u_x t$ for the horizontal distance.

Worked example. A stone is thrown horizontally at 12 m s⁻¹ from a 45 m cliff. $45 = \tfrac{1}{2} \times 9.81 \times t^2$ gives $t = 3.0$ s, so it lands $12 \times 3.03 = 36$ m out. The final vertical velocity is $gt = 29.7$ m s⁻¹, so the speed is $\sqrt{12^2 + 29.7^2} = 32$ m s⁻¹.

At the highest point of a launch at an angle, the vertical velocity is zero and the speed equals the horizontal component.

Friction is a contact force between surfaces that opposes relative motion (or tendency to move). Distinguishing static and dynamic friction is not tested.

Drag (air resistance or fluid resistance) opposes the motion of an object through a fluid. It increases with speed and depends on the shape and area of the object.

Lift is an upward force on an object (such as an aeroplane wing) caused by the fluid flowing over it, perpendicular to the flow.

When an object falls through air, its weight is constant but drag increases with speed:

1. At first drag is zero, so $a = g$.
2. As speed increases, drag increases, so the resultant force (weight − drag) and the acceleration decrease.
3. At terminal speed drag equals weight; the resultant force is zero and the object falls at constant velocity.

Opening a parachute increases the drag suddenly above the weight, so the skydiver decelerates until drag again equals weight at a lower terminal speed.

Without air resistance the path is a symmetrical parabola. With air resistance:

• drag opposes the velocity, so the horizontal velocity decreases
• on the way up, drag has a downward component so the deceleration is greater than g, giving a lower maximum height
• on the way down, drag acts upwards so the acceleration is less than g
• the range is smaller and the path is asymmetrical, with a steeper descent
• the object lands more slowly than it was launched

A vehicle has a driving force from its engine; resistive forces are rolling friction and air resistance. As the speed rises, air resistance rises, so the resultant force falls. At the maximum speed driving force = total resistive force, and the acceleration is zero.

To increase the maximum speed: a larger driving force, or streamlining to reduce drag. Increasing the frontal area or adding a roof box increases drag and lowers the maximum speed.

Worked example. Driving force 1500 N, rolling friction 200 N: at maximum speed drag = 1300 N.