Calculus in physics

Question

Calculus is a powerful tool used in all areas of physics. One of the initial applications areas is classical physics, particularly Newtonian mechanics. An important function is the position function s(t) that measures how far the object is from point 0 at time t. The question below deals with just the position function s(t).

 

Problem Set question:

 

The position function for a falling objects is given by s(t)=−16t2+v0t+s0, where s(t) is the height of the object in feet, v0 is the initial velocity, s0 is the initial height, and t is the time in seconds.

 

A ball is shot upwards from the surface of the earth (s0=0) with an initial velocity of 65 feet per second. What is the maximum height reached by the ball?

 

Round your answer to the nearest integer.

 

The maximum height reached by the ball is ____________.

Answer 1

(Calculus answer)

The position function for the problem is with

v_o = 65 and s_o = 0.

The function

s(t) = - 16 t² +65 t + 0

The first derivative is zero when we reach max height.

s`(t) = - 32 t +65.

-32 t + 65 = 0

t = 65/32 sec.

max height is

s(65/32) =66.01 feet (rounded).

Alternatively, you set s(t) = 0 and solve for t. The max height is the average for the two zeros you found.