Acceleration

• The fundamental relationship for average acceleration is the change in velocity divided by the time interval. This is expressed mathematically as:

$a = \frac{\Delta v}{t} = \frac{v - u}{t}$

• In this formula, $a$ represents acceleration, $v$ is the final velocity, $u$ is the initial velocity, and $t$ is the time taken for the change.

• Positive vs. Negative Acceleration: If the final velocity is greater than the initial velocity, the acceleration is positive (speeding up). If the final velocity is less than the initial velocity, the acceleration is negative, which is commonly referred to as deceleration.

• Uniform Acceleration: This occurs when the velocity of an object changes by the same amount every second. In many introductory physics scenarios, acceleration is assumed to be constant to simplify calculations.

• Gradient as Acceleration: On a velocity-time ($v-t$) graph, the gradient (slope) of the line represents the acceleration. A steeper slope indicates a higher acceleration, while a horizontal line indicates zero acceleration (constant velocity).

• Area as Displacement: The total area under the line of a $v-t$ graph represents the distance travelled or displacement. For constant acceleration, this area usually forms a triangle or a trapezium.

• Curved Lines: If the acceleration is not constant, the $v-t$ graph will be a curve. In such cases, the instantaneous acceleration at a specific point can be found by drawing a tangent to the curve and calculating its gradient.

• Calculating from Change: Use the formula $a = \frac{v-u}{t}$ when you know the starting velocity, ending velocity, and the time duration. Ensure all units are converted to metres and seconds before calculating.

• Uniform Acceleration Equation: When time is unknown, use the equation $v^2 = u^2 + 2as$, where $s$ is the distance travelled. This is particularly useful for calculating braking distances or launch speeds.

• Estimating Real-World Values: For everyday objects, acceleration can be estimated by considering typical speed changes. For example, a car reaching highway speeds in 10 seconds has an acceleration of roughly $3 \text{ m/s}^2$.

Comparison of Motion States

• Acceleration vs. Velocity: Velocity is the speed in a direction; acceleration is how fast that speed or direction is changing. An object can have a very high velocity but zero acceleration if it is moving steadily.

• Negative Acceleration vs. Deceleration: While often used interchangeably, 'negative acceleration' refers to the mathematical sign relative to a coordinate system, whereas 'deceleration' specifically means the object is slowing down.

• Initial Fall: When an object starts falling, the only significant force is weight (gravity), causing it to accelerate downwards at approximately $9.8 \text{ m/s}^2$.

• Increasing Resistance: As the object's speed increases, the upward force of air resistance (friction) also increases. This reduces the resultant force acting on the object, causing the acceleration to decrease.

• Equilibrium: Eventually, the upward air resistance becomes equal to the downward weight. At this point, the resultant force is zero, acceleration stops, and the object falls at a constant speed called terminal velocity.

• Parachutes: Opening a parachute increases surface area, drastically increasing air resistance. This creates a large upward resultant force, causing rapid deceleration until a new, much lower terminal velocity is reached.

• Check the Units: Always ensure acceleration is in $m/s^2$. A common mistake is using $km/h$ for velocity without converting to $m/s$, leading to incorrect acceleration values.

• Identify 'From Rest': If a question states an object starts 'from rest', always set the initial velocity $u = 0$. Similarly, if it 'comes to a stop', the final velocity $v = 0$.

• Graph Interpretation: If asked for distance on a $v-t$ graph, calculate the area. If asked for acceleration, calculate the gradient. Never use $s = v \times t$ if the line is sloped, as that formula only applies to constant velocity.

• Sanity Check: Compare your calculated acceleration to gravity ($9.8 \text{ m/s}^2$). If you calculate a car accelerating at $500 \text{ m/s}^2$, you have likely made a calculation or unit error.