## Linear Motion (Position, Velocity and Acceleration) Involving Integrals

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.

### Practice

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?

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?

$$t=3.04\sec$$

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

### 687 video

video by PatrickJMT

$$t=3.04\sec$$

A particle has acceleration given by $$a(t)=12t-4$$ with initial velocity $$-10$$ and initial position $$0$$. Find its velocity and position functions.

Problem Statement

A particle has acceleration given by $$a(t)=12t-4$$ with initial velocity $$-10$$ and initial position $$0$$. Find its velocity and position functions.

$$v(t)=6t^2-4t-10$$
$$x(t)=2t^3-2t^2-10t$$

Problem Statement

A particle has acceleration given by $$a(t)=12t-4$$ with initial velocity $$-10$$ and initial position $$0$$. Find its velocity and position functions.

Solution

### 694 video

video by Krista King Math

$$v(t)=6t^2-4t-10$$
$$x(t)=2t^3-2t^2-10t$$

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?

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?

max height $$144~ft$$; $$t=6~secs$$

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

### 695 video

video by Krista King Math

max height $$144~ft$$; $$t=6~secs$$

A car braked with a constant deceleration of 16 ft/s2, producing skid marks measuring 200ft before coming to a stop. How fast was the car traveling when the brakes were first applied?

Problem Statement

A car braked with a constant deceleration of 16 ft/s2, producing skid marks measuring 200ft before coming to a stop. How fast was the car traveling when the brakes were first applied?

$$v=80 ft/\sec$$

Problem Statement

A car braked with a constant deceleration of 16 ft/s2, producing skid marks measuring 200ft before coming to a stop. How fast was the car traveling when the brakes were first applied?

Solution

### 688 video

video by PatrickJMT

$$v=80 ft/\sec$$

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