Mathematics

Mathematics

Problem

A car moves along a straight road so that its displacement from a fixed milestone at time tt seconds is given by s(t)=5t3−18t2+12t+20 metres.s(t)=5t^{3}-18t^{2}+12t+20 \text{ metres}. Answer the following questions based on this model. a. (i) Find an expression for the instantaneous velocity v(t)v(t) of the car. b. (ii) Calculate the average velocity of the car during the time interval t=2 st=2\,\text{s} to t=4 st=4\,\text{s}. c. (iii) Determine the instantaneous velocity at t=4 st=4\,\text{s}. d. (iv) Compare the answers obtained in parts (ii) and (iii). What does this tell you about the motion of the car in this interval? e. (v) State the physical units of the instantaneous velocity function v(t)v(t).
Intuitive explanation

Displacement, velocity, and acceleration are connected through derivatives—each one tells you how the previous quantity is changing. To answer questions about motion, you'll need to find s′(t)s'(t) for velocity and s′′(t)s''(t) for acceleration, then use algebra and calculus to locate key moments like when the car stops or changes direction. This is about translating a position formula into a complete picture of the car's motion.

Intuitive explanation

Displacement, velocity, and acceleration are connected through derivatives—each one tells you how the previous quantity is changing. To answer questions about motion, you'll need to find s′(t)s'(t) for velocity and s′′(t)s''(t) for acceleration, then use algebra and calculus to locate key moments like when the car stops or changes direction. This is about translating a position formula into a complete picture of the car's motion.

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