Give the positions s=f(t) of a body moving on a coordinate line, with s in meters and t in secon

Give the positions s=f(t) of a body moving on a coordinate...

Key Concepts

Change of Direction

A change of direction in one-dimensional motion occurs when the velocity of the object changes its sign, which means that the object transitions from moving in one direction to moving in the opposite direction.

Differentiation in Kinematics

Differentiation is a fundamental mathematical tool in kinematics that involves computing derivatives of functions. The first derivative of the position function yields the velocity, while the second derivative provides the acceleration, offering insights into the motion characteristics of an object.

Acceleration

Acceleration is the rate at which velocity changes with time. It is obtained by differentiating the velocity function with respect to time (or differentiating the position function twice) and indicates how quickly an object speeds up or slows down.

Average Velocity

Average velocity is defined as the displacement divided by the time interval over which the displacement occurred. It provides a measure of the overall rate of change of position and is a vector quantity, indicating both magnitude and direction.

Displacement

Displacement refers to the overall change in position of an object from its initial location to its final location within a specified time interval. It is a vector quantity that considers only the starting and ending points, not the path taken.

Speed

Speed is a scalar quantity that represents how fast an object is moving, without regard to direction. It is the magnitude of the velocity and can vary at different instants during the motion.

Transcript

00:01 Problem here we are given this function representing our position.

00:05 So t to the fourth over four minus t squared plus t squared on the interval from zero to three seconds.

00:15 And so looking at part a, we're being asked, what's our displacement? first we're being asked what's our just displacement? delta x usually however in this case it's definitely s right.

00:26 To find this, this is basically the final position minus the initial position, and according to our timeline.

00:35 So basically s of 3 minus s of 0.

00:39 So plugging in the values of 3 and 0, so for 3 we get 81 or 4, that's 3 to the 4th, minus 27 plus 9.

00:49 So this becomes torts to 8104 minus 18.

00:54 So because we want to find a common denominator, right, we do this to convert it and this becomes 81 over 4 minus 72 over 4.

01:07 So this is 9 over 4.

01:09 So we know that s final is 9 over 4.

01:14 And then as for s initial, because there are ts on every side of this on every part of the of our polymenomial, we know that this is equal 0.

01:24 So our total displacement is 9 4ths.

01:31 All right, now moving on to the second part of part a, we're being asked, okay, so now that we have our displacement, what's our average velocity? so to find the average velocity, this is basically our change in position, so our displacement over the change in time.

01:46 So we already got our change in position right here.

01:53 We get 9 4ths.

01:55 And then we want to divide this by our time.

01:58 So 3, right? so when this happens, we get that our average velocity is 3 fourth meters per second.

02:10 Now moving into part b, we're being asked.

02:13 Okay, so you're on the interval, 0 to 3.

02:15 Now, what's the speed on each of these endpoints, right? and what's the acceleration? so to find the speed, the speed is the absolute value of our velocity.

02:27 So how do we find the velocity, you might ask? well.

02:30 To find the velocity, you would take the derivative of our position.

02:34 So our position is this right here, right? so taking the derivative, 4 counts of 4, we get, first of all, t cubed, right? but then negative 3 t squared, and then plus 2t.

02:49 So now that we have this here, we can plug in that values of 0 and 3.

02:53 All right, so for a 0, because there's a t on every part of this polynomial, we get that this is 0.

03:01 Meters per second.

03:03 Now for three, we'd have to calculate this.

03:05 So this would be 27 minus 27, right? the 27 is cancel out plus six.

03:11 Therefore it would be six meters per second for three.

03:15 As for acceleration, well, same thing with what we did to get velocity to find the acceleration.

03:21 This is b prime of t.

03:23 So the derivative of velocity would be acceleration.

03:28 So this would be 3t squared minus 6 t plus 2.

03:33 So plugging in our values, say we plug in 0 right at this endpoint, get that at 0, it's 2 meters per second squared.

03:43 And then we plug it 3.