Differential calculus, Physics problem

In summary, the projectile fired with a velocity of 400m/s reaches a maximum altitude of 2500m and hits the ground after 25 seconds at a velocity of 400m/s. The acceleration of the projectile at any time is -32m/s^2.
  • #1
menco
43
0

Homework Statement


A projectile is fired straight upward with a velosict of 400m/s. From physics, its distance above the ground after t seconds is: s(t) = -16t^2 + 400t

A Find the time and velocity at which the projectile hits the ground

B Find the maximum altitude achieved by the projectile

C Find the acceleration at any time t.

Homework Equations


s(t) = -16t^2 + 400t


The Attempt at a Solution



A The surface s(t) = 0

0 = -16t^2+400t
16t^2-400 = 0
16t(t-25) = 0
16t = 0 t-25 = 0
t = 0 t = 25

v(t) = s'(t) = -32t+400
v(25) = -32(25)+400
= -400

The projectile hits the ground after 25 seconds at a velocity of 400m/s

B s'(t) = v(t) = -32t+400
0 = -32t+400
32t = 400
t = 12.5
s(12.5) = -16(12.5)^2+400(12.5)
= 2500m

The projectile reaches a max altitude of 2500m

C a(t) = v'(t) = -32m/s^2

The acceleration at anytime is 32m/s^2
 
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  • #2
Huh, I wonder if the original problem had a typo, because the acceleration here is calculated at -32 m/s^2, but on Earth g ≈ -32 ft/s^2. In metric units, g ≈ -9.8 m/s^2.

Besides this curiosity, I see nothing wrong with your work, except for a small typo on the 2nd line of A:
16t^2-400 = 0
should be
16t^2-400t = 0.
 
  • #3
Well hopefully their answers have a typo as well or I will be way off. Good to know I got the calculations right tho, Thanks for the help
 

1. What is differential calculus?

Differential calculus is a branch of mathematics that deals with the study of rates of change. It involves finding the slope of a curve at a specific point, known as the derivative, and using this information to solve various problems in physics, engineering, and other fields.

2. How is differential calculus used in physics?

Differential calculus is extensively used in physics to analyze and model the motion of objects. It is used to determine the velocity and acceleration of an object at a given point, which helps in understanding the forces acting on the object and predicting its future motion.

3. What is the difference between differential calculus and integral calculus?

Integral calculus is the inverse process of differential calculus. While differential calculus deals with finding the derivative of a function, integral calculus deals with finding the area under a curve. It is used to solve problems related to finding the total displacement, distance, or area.

4. How can I improve my understanding of differential calculus?

To improve your understanding of differential calculus, it is important to have a strong foundation in algebra, trigonometry, and basic calculus concepts. Practicing problems and visualizing the concepts through graphs and diagrams can also help in better understanding.

5. Can differential calculus be applied in real-world scenarios?

Yes, differential calculus has numerous applications in the real world, especially in fields such as physics, engineering, economics, and statistics. It is used to solve problems related to optimization, motion, and change, making it a valuable tool in many industries and fields of study.

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