3.6 Lab -- Derivatives as Rates of Change Student: Student: HlO Calculus I 1.) O

Question

3.6 Lab -- Derivatives as Rates of Change Student: Student: HlO Calculus I 1.) On Earth's moon, a feather will fall to the ground at the same rate as a heavy stone. Suppose a feather is dropped from a height of 40 m above the surface of the moon. Then its height s (in meters) above the ground after t seconds is s 40 0.8t, Determine the velocity and acceleration of the feather the moment it strikes the surface of the moon. Round your answers to the nearest tenth, and include appropriate units for full credit. 5-40-0.8 S46-0.8 ?? Velocity: 10% Acceleration:1-6 2.) A stone is thrown vertically into the air at an initial velocity of 96 ft/s. On Mars, the height s (in feet) of the stone above the ground after t seconds is s 96t-6t2 and on Earth it is s-96t-16t. How much higher will the stone travel on Mars than on Earth? Answer:

Explanation / Answer

The height of the falling object (whether feather or iron ball) is given by
s(t) = 40 - 0.8t²

The velocity of the falling object is the derivative of the height function:
v(t) = -1.6t

The acceleration of the falling object is the derivative of the velocity function:
a(t) = -1.6

Solution:
Acceleration: Since the acceleration is constant and unchanging with time, the acceleration at the instant of impact is the constant value, -1.6 m/sec²

Velocity: The velocity at impact is the velocity at the instant of time that the height becomes zero.
s(t) = 0 = 40-0.8t²
t² = 40/0.8 = 50
t = ?50 = 5?2

v(5?2) = -1.6*5?2 = -8?2 m/sec

2) The motion functions for the movement of the stone on Mars and on Earth are given in the question:

sm(t) = 96t - 6t2 ; se(t) = 96t - 16t2  

The stone reaches its highest point when its velocity becomes zero, so we need to calculate the velocityfunction in these two cases

vm(t) = 96 -12t , ve(t)= 96-32

Let vm(t) = 0, andve(t) = 0, we can find out the time at which the stone reaches its highest point onMars and on Earth, respectively. Those two times are:

tm= 8s,

te = 3s.

Then we can calculate the highest point of the stone by substituting the two times into corresponding motion functions.

Hm=sm (tm) = 96·8-6·82= 768-384 = 384

He=se(te) = 96·3-16·32= 288-144 = 144

Hm-He= 384-144 = 240

So the ball will travel 240 feet higher on Mars than on Earth

Related Questions

Navigate

• Browse (All)
• Browse 3
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.