When using calculus for a useful application, the equations and subsequent derivatives usually mean something or describe something. On this page, we discuss the situation when a function represents the position of an object, in two dimension motion, vertically, horizontally or a combination. Calculus, including derivatives and integrals, are the only requirements to understand this material. For a more advanced discussion, see the differential equations page on projectile motion. For a discussion involving only derivatives, see this page. We assume you know the material on the derivatives linear motion page for the following discussion.
Integration is required when you are given a velocity equation and you are asked to come up with the position function or when you are given acceleration and asked to come up with the velocity and/or the position function. So, essentially, it looks like this.
derivative  position → velocity → acceleration 
integral  position ← velocity ← acceleration 
Of course, with integration, an unknown constant is introduced. So, you will often be given a value that allows you to determine the unknown constant. This value is often called an initial condition. If the value is a position, it may be called an initial position. If it is a velocity, it may be called an intial velocity. You will find the word initial used a lot since most of the time, the value at time \(t=0\) is given.
Let's work some practice problems to get an idea how this works.
Problem Statement 

A penny is thrown downward from a 300 foot tall tower with an initial velocity of 50 ft/sec. When does the penny hit the ground?
Final Answer 

Problem Statement 

A penny is thrown downward from a 300 foot tall tower with an initial velocity of 50 ft/sec. When does the penny hit the ground?
Solution 

video by PatrickJMT 

Final Answer 

\(t=3.04\sec\) 
close solution

Problem Statement 

A particle has acceleration given by \(a(t)=12t4\) with initial velocity \(10\) and initial position \(0\). Find its velocity and position functions.
Final Answer 

Problem Statement 

A particle has acceleration given by \(a(t)=12t4\) with initial velocity \(10\) and initial position \(0\). Find its velocity and position functions.
Solution 

video by Krista King Math 

Final Answer 

\(v(t)=6t^24t10\) 
close solution

Problem Statement 

A ball is thrown straight upward from the ground with initial velocity 96 ft/s. How high does the ball rise and how long does it remain aloft?
Hint 

If you are having trouble getting started on this problem, try using the equation \(v(t)=gt+v_0\) for velocity where g is the acceleration due to gravity. The negative sign indicates up as the positive direction.
Problem Statement 

A ball is thrown straight upward from the ground with initial velocity 96 ft/s. How high does the ball rise and how long does it remain aloft?
Final Answer 

Problem Statement 

A ball is thrown straight upward from the ground with initial velocity 96 ft/s. How high does the ball rise and how long does it remain aloft?
Hint 

If you are having trouble getting started on this problem, try using the equation \(v(t)=gt+v_0\) for velocity where g is the acceleration due to gravity. The negative sign indicates up as the positive direction.
Solution 

video by Krista King Math 

Final Answer 

max height \(144~ft\); \(t=6~secs\) 
close solution

Problem Statement 

A car braked with a constant deceleration of 16 ft/s^{2}, producing skid marks measuring 200ft before coming to a stop. How fast was the car traveling when the brakes were first applied?
Final Answer 

Problem Statement 

A car braked with a constant deceleration of 16 ft/s^{2}, producing skid marks measuring 200ft before coming to a stop. How fast was the car traveling when the brakes were first applied?
Solution 

video by PatrickJMT 

Final Answer 

\(v=80 ft/\sec\) 
close solution

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