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Acceleration

This guide explains the relationship between acceleration, velocity, and position in motion. It covers the concepts of constant velocity and changing velocity, demonstrating how they relate to straight-line graphs. By using examples, such as a jet plane's takeoff speed and average acceleration, readers will gain insight into the mathematical expressions of motion. The guide also discusses important calculus concepts like derivatives, highlighting how instantaneous velocity and acceleration can be derived. Practical applications are provided to enhance understanding.

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Acceleration

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  1. Acceleration

  2. x v t t Graphs to Functions • A simple graph of constant velocity corresponds to a position graph that is a straight line. • The functional form of the position is • This is a straight line and only applies to straight lines. x0 v0

  3. Changing Velocity • In more complicated motion the velocity is not constant. • We can express a time rate of change for velocity just as for position, v = v2 - v1. • The acceleration is the time rate of change of velocity: a = v / t.

  4. Average Acceleration Example problem • A jet plane has a takeoff speed of 250 km/h. If the plane starts from rest, and lifts off in 1.2 min what is the average acceleration? • a = v / t = [(250 km/h) / (1.2 min)] * (60 min/h) • a = 1.25 x 104 km/h2 • Why is this so large? Is it reasonable? • Does the jet accelerate for an hour?

  5. Instantaneous velocity is defined by a derivative. Instantaneous acceleration is also defined by a derivative. v P2 P1 P3 t P4 Instantaneous Acceleration

  6. Second Derivative • The acceleration is the derivative of velocity with respect to time. • The velocity is the derivative of position with respect to time. • This makes the acceleration the second derivative of position with respect to time.

  7. v a t t Constant Acceleration • Constant velocity gives a straight line position graph. • Constant acceleration gives a straight line velocity graph. • The functional form of the velocity is v0 a0

  8. For constant acceleration the average acceleration equals the instantaneous acceleration. Since the average of a line of constant slope is the midpoint: v t Acceleration and Position a0(½t) + v0 ½t v0

  9. Algebra can be used to eliminate time from the equation. This gives a relation between acceleration, velocity and position. For an initial or final velocity of zero. This becomes x = v2 / 2a v2 = 2 a x Acceleration Relationships from

  10. Take Off Example problem • A jet plane has a takeoff speed of 250 km/h. If the plane starts from rest, and has a constant acceleration of 1.25 x 104 km/h2, what is the length of the runway? • x = v2 / 2a = (250 km/h)2 / (2.5 x 104 km/h2) • x = 2.5 km • Is this reasonable? next

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