Derivatives and Physics Word Problems

1The equation of a rectilinear movement is: d(t) = t³ − 27t. At what moment is the velocity zero? Also, what is the acceleration at this moment?

2What is the speed that a vehicle is travelling according to the equation d(t) = 2 − 3t2 at the fifth second of its journey? In this intance, space is measured in meters and time in seconds.

3Due to bad environmental conditions, a colony of a million bacteria does not reproduce during the first two months of a study. The function that represents the population of the colony during the entire study (time is represented in months) is given by:

Function

1. Verify that the population is a continuous function of time.

2. Calculate the average rate of change of the population during the interval [0, 2] and [0, 4].

3. Calculate the instantaneous rate of change at t = 4.

4The growth of a bacterial population is represented by the function p(t) = 5,000 + 1,000t², where t is the time measured in hours. Determine:

1. The average growth rate.

2. The instantaneous rate of growth.

3. The instantaneous growth rate at t0 = 10 hours.

5The equation of a circular motion is: φ(t) = ½t². What is the angular velocity and the acceleration at the seven second mark?

6A man is 2000 m from the base of a tower and is launching a rocket in the direction of the same tower. When the rocket takes off the change in the angle between the flight path and the land is represented by Φ(t) according to time. Knowing that Φ'(t) = Π/3, determine:

1. The height of the rocket when Φ = Π/3 radians.

2. The velocity of the rocket when Φ = Π / 3 radians?

7Gas is pumped into a spherical tank at 6 m3/min. If the pressure remains constant, at what velocity does the size of the radius change when the diameter is 120 cm?


1

The equation of a rectilinear movement is: d(t) = t³ − 27t. At what moment is the velocity zero? Also, what is the acceleration at this moment?

v(t) = d′t) = 3t² − 27 3t² − 27 = 0t = ± 3

a(t) = d′'(t) = 6ta(−3) = −18a(3) = 18


2

What is the speed that a vehicle is travelling according to the equation d(t) = 2 − 3t² at the fifth second of its journey? In this intance, space is measured in meters and time in seconds.

Velocity Function

Physics Solution


3

Due to bad environmental conditions, a colony of a million bacteria does not reproduce during the first two months of a study. The function that represents the population of the colony during the entire study (time is represented in months) is given by:

Function

1. Verify that the population is a continuous function of time.

Derivative Operations

Derivative Operations

Derivative Operations

2. Calculate the average rate of change of the population during the interval [0, 2] and [0, 4].

Derivative Operations

Derivative Operations

3. Calculate the instantaneous rate of change at t = 4.

Derivative Operations

Physics Solution


4

The growth of a bacterial population is represented by the function p(t) = 5,000 + 1,000t², where t is the time measured in hours. Determine:

1. The average growth rate.

Growth Rate Function

Derivative Operations

2. The instantaneous rate of growth.

Derivative Operations

3. The instantaneous growth rate at t0 = 10 hours.

Physics Solution


5

The equation of a circular motion is: φ(t) = ½t². What is the angular velocity and the acceleration at the seven second mark?

ω(t)= φ′(t)= t ω = 7

α(t)= φ′′ (t)= 1 α = 1


6

A man is 2,000 m from the base of a tower and is launching a rocket in the direction of the same tower. When the rocket takes off the change in the angle between the flight path and the land is represented by Φ(t) according to time. Knowing that Φ'(t) = Π/3, determine:

1. The height of the rocket when Φ = Π/3 radians.

Physics Diagram

Derivative Operations

2. The velocity of the rocket when Φ = Π/3 radians?

Derivative Operations

Derivative Operations

Physics Solution


7

Gas is pumped into a spherical tank at 6 m3/min. If the pressure remains constant, at what velocity does the size of the radius change when the diameter is 120 cm?

Velocity Function

Derivative Operations

Derivative Operations

Derivative Operations

Physics Solution




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