The acceleration function (in m / s^2) and the initial velocity are given for a particle moving

The acceleration function (in m / s^2) and the initial...

Key Concepts

Definite Integration

Definite integration is used in kinematics to calculate the net change in quantities over a specific interval. For example, by integrating the velocity function over a given time period, one can determine the total displacement or distance traveled by a moving object.

Kinematics

Kinematics is the branch of physics that describes the motion of objects without considering the forces that cause the motion. It involves analyzing variables such as displacement, velocity, and acceleration, and their interrelationships through calculus-based methods.

Distance Traveled vs Displacement

While displacement refers to the net change in position and can be affected by direction, distance traveled is the total length of the path taken by an object. Calculating distance often requires integrating the absolute value of the velocity if the object changes direction during its motion.

Integration

Integration is the mathematical process used to find antiderivatives. In kinematics, integrating the acceleration function yields the velocity function, and further integrating the velocity function determines the displacement of an object over time. This process also involves applying definite integrals to compute changes over specified time intervals.

Initial Conditions

Initial conditions, such as the given initial velocity, are crucial in solving initial value problems. They allow for the determination of constants of integration when recovering original functions from their derivatives, ensuring that the solution accurately reflects the system's state at the starting time.

Transcript

0:00 Hello.

00:02 Today we're going to solve a simple, simple integration problem.

00:06 So let's get started.

00:09 So an acceleration function, we're going to define it as a times t is going to be defined as t plus four, where t is the time.

00:25 What we're going to do is first we are going to find the velocity at time t.

00:31 So the velocity at time t at time t why is it t well it means that it's an arbitrary time so given any amount of time right we're going to put we're going to have a function where we're going to tell what the velocity is in respect to that point in time so we're not giving an exact amount which is giving out a formula and then v is going to be the distant travel traveled during a time interval from zero to 10 all -inclusive so from zero to 10 and let me set the initial condition at t equals 0 v equals 5 so let's get started so let's first refresh our knowledge what is acceleration well acceleration is the rate of change of velocity in respect to time right so that's going to be delta t i mean delta v over delta t well in the case of calculus it's going to be an instant change meaning it's going to be the derivative of v in respect to t and so therefore manipulating this function what we're going to have is that when we multiply both sides by d of t we're going to have that so we're going to have dv equals a times d of t if we were to integrate this then the differential and the integral they cancel out.

03:03 We're going to have v as a function of t equals to the antiderivative of a times d of t.

03:12 So basically to find the velocity function, all we have to do is integrate the acceleration function.

03:20 So the integral of a of t in respect to d of t is going to be the integral of t plus four.

03:32 Times d t so using the power roll we can conclude that the answer derivative of t is going to be t squared divided by two plus 40 plus a constant c and that is going to be our velocity function there it is so how do we solve for this constant c here well remember that we had this initial property so what we can do is at every point of t we see, we're gonna plug in t equals zero.

04:15 And then the function is going to output a five.

04:19 So when t is zero, then the velocity is gonna be five.

04:25 Five equals zero squared divided by two plus four times zero equals c.

04:32 So zero squared is zero, zero divided by two is zero.

04:36 Four times zero is zero.

04:38 And so therefore, c, is equals to 5.

04:42 And now we can substitute the c to 5.

04:49 And there we have it.

04:50 This is our velocity function.

04:53 So whenever we have t, we can just plug it into the function and we're going to get the velocity.

05:00 So that's the first part.

05:03 Now the second part, we're going to find the distance that we have traveled.