Question 1

Paper 2Hard8 marks

During a test, a small cart moves along a straight track. For 0≤t≤80\leq t\leq 8 its acceleration is modelled by

dvdt=−0.072t2+0.610t−1.05m s−2,\frac{dv}{dt}=-0.072t^2+0.610t-1.05\quad\text{m s}^{-2},

where tt is in seconds and vv is in m s−1\text{m s}^{-1} The cart’s velocity at t=0t=0 is 1.02 m s−11.02\text{ m s}^{-1}

(a)

Show that

v(t)=−0.024t3+0.305t2−1.05t+1.02.v(t)=-0.024t^3+0.305t^2-1.05t+1.02.
[2]
(b)

Use your GDC to find all values of tt for which the cart is momentarily at rest. Give your answers to three significant figures.

[2]
(c)

A technician claims that, since the cart starts moving in the positive direction, its forward distance over 0≤t≤80\leq t\leq 8 is ∫08v(t) dt\int_0^8 v(t)\,dt

Assess this claim, giving a reason. Then determine the total distance travelled while the cart is moving in the positive direction. Use your results from part (b), write the definite-integral expression(s) before evaluating them with your GDC, and give the distance to three significant figures.

[4]

More AI 5.2 Integration and kinematics questions