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IB DP Maths AA SL Study Notes

5.6.2 Acceleration

Definition of Acceleration

Acceleration is the rate at which an object's velocity changes over time. It's a measure of how quickly an object speeds up, slows down, or changes direction.

  • Mathematical Representation: If we denote the velocity of an object at time t as v(t), then the acceleration, represented as a(t), is the derivative of v(t) concerning t. In mathematical terms, a(t) = dv(t)/dt.

Relationship with Velocity

The relationship between acceleration and velocity is both profound and multifaceted:

  • Directional Implications: The sign of acceleration provides insights into the direction of velocity change.
    • Positive Acceleration: When a(t) > 0, it signifies an increase in the object's velocity.
    • Negative Acceleration: Conversely, a(t) < 0 indicates a decrease in velocity.
  • Magnitude and Rate: The absolute value of acceleration depicts the rate at which velocity changes, regardless of direction.
  • Stationary Phase: An object with zero acceleration has a constant velocity, implying it moves at a uniform speed.

Delving Deeper: Acceleration in Various Scenarios

Constant Acceleration

When an object's acceleration remains unchanged over time, it's said to have constant acceleration. This is commonly observed in free-falling objects under the influence of gravity, excluding air resistance.

Example: A stone dropped from a height experiences constant acceleration due to gravity, which is approximately 9.81 m/s2 on Earth's surface.

Variable Acceleration

In many real-world scenarios, objects don't have a constant acceleration. Instead, their acceleration varies due to factors like friction, air resistance, or external forces.

Example: A car accelerating from a stationary position might experience variable acceleration due to factors like engine performance, road conditions, and tyre friction.

Analysing Motion through Graphs

Graphical representations offer invaluable insights into an object's motion:

  • Velocity-Time Graph: The gradient or slope of a velocity-time graph at any given point provides the acceleration. The area beneath the curve represents the object's displacement over a time interval.
  • Acceleration-Time Graph: The area under an acceleration-time graph over a specific time interval gives the change in velocity during that period.

Example: Consider an acceleration-time graph represented by a straight line passing through the origin with a gradient of 3 m/s3. To determine the change in velocity over 4 seconds, one would calculate the area under the graph for that duration. Using the formula for the area of a triangle (0.5 x base x height), the change in velocity is 0.5 x 4 s x 3 m/s3 x 4 s, which equals 24 m/s.

Real-World Implications of Acceleration

Automotive Industry

In vehicle design and performance testing, understanding acceleration is paramount. It influences engine efficiency, braking mechanisms, and safety features. For instance, rapid acceleration capabilities can be a selling point for sports cars, while efficient deceleration is crucial for safety.

Aerospace and Space Missions

For rockets and spacecraft, acceleration determines thrust requirements, fuel consumption rates, and trajectory optimisation. Successful space missions hinge on precise calculations of acceleration at various stages of the journey.

Athletic Training

In sports, especially track events, an athlete's acceleration can be a game-changer. Sprinters, for instance, train rigorously to achieve maximum acceleration in minimal time, giving them a competitive edge right off the starting block.

Everyday Life

Even in our daily lives, acceleration plays a role. From the feel of being pushed back in our seats when a car accelerates to the sensation of weightlessness in a descending lift, these are all manifestations of acceleration.

FAQ

Air resistance, or drag, opposes the motion of an object moving through the air. For free-falling objects, air resistance acts upwards, opposing the downward force of gravity. Initially, when an object starts to fall, air resistance is less than the gravitational force, so the object accelerates. However, as its speed increases, air resistance also increases. Eventually, a point is reached where air resistance equals the gravitational force. At this point, the object stops accelerating and falls at a constant velocity, known as terminal velocity.

In a vacuum, there's no air resistance to oppose the motion of falling objects. As a result, all objects, regardless of their mass or shape, experience the same gravitational force and hence the same acceleration. This means they fall at the same rate. This principle was famously demonstrated by Galileo when he dropped two spheres of different masses from the Leaning Tower of Pisa and observed that they hit the ground simultaneously. In the absence of air resistance, the only force acting on a free-falling object is gravity, making their rates of fall identical.

No, if an object has a constant velocity, its speed and direction remain unchanged. Since acceleration is the rate of change of velocity, a constant velocity implies zero acceleration. An object can only be said to be accelerating if there's a change in its velocity, either in magnitude, direction, or both.

Negative acceleration simply means that the acceleration has a negative value, which can indicate a decrease in velocity. Deceleration, however, specifically refers to a reduction in speed or the slowing down of an object. All decelerations are negative accelerations, but not all negative accelerations are decelerations. For instance, if an object is moving in the negative direction and its speed increases, it has a negative acceleration but is not decelerating.

Instantaneous acceleration refers to the acceleration of an object at a specific moment in time. It's the rate of change of velocity at that exact instant. On the other hand, average acceleration is calculated over a defined time interval. It's the change in velocity divided by the time taken for that change. While instantaneous acceleration gives a precise value at a particular moment, average acceleration provides an overall view of how acceleration has behaved over a period.

Practice Questions

A car's velocity, v(t), in m/s, as a function of time, t, in seconds, is given by the equation v(t) = t^2 - 4t + 3. Determine the time at which the car's acceleration is zero.

To find the time at which the car's acceleration is zero, we first need to determine the car's acceleration as a function of time. The acceleration, a(t), is the derivative of the velocity function, v(t). Differentiating v(t) = t2 - 4t + 3 with respect to t, we get a(t) = 2t - 4. Setting this equal to zero, 2t - 4 = 0, we find t = 2 seconds. Therefore, the car's acceleration is zero at t = 2 seconds.

Using the same velocity function v(t) = t^2 - 4t + 3, determine the car's acceleration at t = 3 seconds.

To determine the car's acceleration at t = 3 seconds, we use the acceleration function derived earlier, a(t) = 2t - 4. Substituting t = 3 into this equation, we get a(3) = 2(3) - 4 = 2 m/s2. Thus, the car's acceleration at t = 3 seconds is 2 m/s2.

Dr Rahil Sachak-Patwa avatar
Written by: Dr Rahil Sachak-Patwa
LinkedIn
Oxford University - PhD Mathematics

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.

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