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Acceleration

Acceleration. 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. x 0. v 0. Changing Velocity.

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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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