Application Problems on Derivatives

Problem 1 :

The distance x metres described by a car in time t seconds is given by :

x = 3t³ – 2t² + 4t – 1

Determine the velocity and acceleration when

(i) t = 0 and

(ii) t = 1.5 s

Solution :

Problem 2 :

The angular displacement θ radians of a fly wheel varies with time t seconds and follows the equation

θ = 9t² – 2t³

Determine
(i) the angular velocity and acceleration of the fly wheel when time t = 1 second and
(ii) the time when the angular acceleration is zero.

Solution :

Problem 3 :

Supplies are dropped from an helicopter and distance fallen in time t seconds is given by

x = (½)gt² 

where g = 9.8 m/sec². Determine the velocity and acceleration of the supplies after it has fallen for 2 seconds.

Solution :

Problem 4 :

A missile fired from ground level rises x meters vertically upwards in t seconds and

Find :

(i) the initial velocity of the missile,

(ii) the time when the height of the missile is a maximum

(iii) the maximum height reached and

(iv) the velocity with which the missile strikes the ground

Solution :

Problem 5 :

A boy, who is standing on a pole of height 14.7 m throws a stone vertically upwards. It moves in a vertical line slightly away from the pole and falls on the ground. Its equation of motion in meters and seconds is

x = 9.8t – 4.9t²

(i) Find the time taken for upward and downward motions.

(ii) Also find the maximum height reached by the stone from the ground.

Solution :

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