Variable Acceleration

Velocity What is the gradient of a displacement time graph? How do we normally write gradients?

Acceleration What is the gradient of a velocity time graph equal to? How would we write this?

How do we use this? For the first 4 seconds, the displacement, in metres, of a sports car from its initial position is given by: s = 12t2 – t3 How can we find an expression for the velocity? v = ds/dt v = 24t – 3t2 What about acceleration? a = dv/dt a = 24 – 6t

Solving problems Using the expressions found on the previous slide, find: The initial velocity The velocity after 4 seconds The acceleration after 4 seconds The national speed limit in Great Britain is 70mph. At the end of 4 seconds, would the driver be breaking the speed limit?

Summary Given the displacement as an expression involving time: The velocity can be found by differentiating the displacement The acceleration can be found by differentiating the velocity

Variable Acceleration Part 2

Displacement What is the area under a velocity time graph equal to? How do we find areas under curved graphs? This result also follows for finding the velocity from the acceleration

How do we use this? The acceleration of a particle, in ms-2, at time t seconds is given by a = 6 – 2t When t=0, the particle is at rest at a point 4m from the origin O Find expressions for the velocity and the displacement in terms of t Find when the particle is next at rest, and its displacement from O at that time

Summary Given the acceleration as an expression involving time: An expression for velocity can be found by integrating the acceleration (don’t forget about +c) An expression for displacement can be found by integrating the velocity (don’t forget about +c)