general motion – using calculus
recap
Use the graph to find approximate answers to the questions above.
What mathematical techniques would be required to find the exact answers?
video (be in slide show)
The particle is still speeding up (ie accelerating) not slowing down up to the maximum
Using differentiation
A particle P moves along the x-axis. At time t seconds the velocity of P is v ms–1 in the positive x-direction, where v = 3t2 – 4t + 3.
By drawing a tangent to the curve at t = 1, estimate the acceleration of the P at t = 1 second.
By differentiating v(t), find the exact value of the acceleration at t = 1.
From the graph, estimate the time at which P has minimum velocity.
Use differentiation to find an expression for the acceleration of P.
Find the exact time at which P has minimum velocity.
Find an expression for the acceleration in 0<t<4.
Find the time at which the velocity of P is greatest.
Hence find the greatest speed of P.
What does the graph of Y2 show?
What does the graph of Y3 show?
How do the graphs show the relationships between displacement, velocity and acceleration?
4.
5.
6.
7.
The graph shows the motion of P along a straight line.
Write a description of the motion. When is the velocity a maximum in the positive direction? What happens at t = 8 seconds? What happens for t > 8 seconds?
Use calculus to answer part a).
Why is the model unrealistic?
With reference to part b) what does the shaded area(s) represent?
Find an expression for the displacement of P. Explain why the constant of integration is zero.
Explain why you will set the displacement equal to zero.
Find the time taken for P to return to O.
Using integration
video
1
2
3
4
5
6
7
8
9
video
Some exam questions