The force on a particle is given by 𝐟(t)=(cost) 𝐢+(sint) 𝐣 . The particle is located at point (c

The force on a particle is given by 𝐟(t)=(cost) 𝐢+(sint) 𝐣 ....

Key Concepts

Trajectory Calculation of a Particle

Determining the path of a particle involves combining the integration of acceleration and velocity with the appropriate initial conditions to obtain the position vector as a function of time. The resulting trajectory describes the spatial path the particle follows under the influence of the applied force.

Initial Conditions in Differential Equations

When solving differential equations that govern motion, initial conditions, such as the starting position and velocity, are used to determine the constants of integration that arise during the integration process. These conditions ensure that the obtained solution accurately reflects the behavior of the system from a known state.

Newton's Second Law

Newton's Second Law (F = m·a) is a fundamental principle of classical mechanics stating that the force acting on an object is equal to its mass multiplied by its acceleration. This law provides the basis for predicting the motion of objects by relating the net force acting on them to the rate of change of their velocity.

Vector Integration in Kinematics

In kinematics, acceleration, velocity, and displacement are vector quantities. To find the velocity from a given acceleration, one integrates the acceleration vector with respect to time; similarly, integrating the velocity vector with respect to time yields the displacement or the path of the particle. This process captures the direction and magnitude of motion in a multi-dimensional space.

Transcript

00:01 They tell us that the force of a particle is given by this equation f of t here.

00:07 And they want us to find the position function based off these two initial conditions that we have over here on the left.

00:17 So they pretty much give us the hint for this that if we recall that force is equal to mass times acceleration, we can use this to help us find our position.

00:33 So let's see.

00:34 Well, we know that if we integrate acceleration with respect to time, we should get velocity.

00:42 And if we integrate velocity with respect to time, we should get our position function.

00:49 So if we take our force function here and divide each side by m, we'll end up with our acceleration.

00:56 So the first thing i'm just going to do is rewrite this.

00:59 In the bracket notation, just because i think it looks a little bit prettier.

01:03 So the i component is going to be the x, and the j component is going to be the y.

01:11 So we end up with this here.

01:14 So if we want to find acceleration, so a of t, well this is going to be f of t divided by m.

01:22 So we go ahead and divide this by m.

01:25 So we'd have 1 over m out here, cosine t.

01:31 Sine t.

01:33 And then remember when we multiply or divide a vector by a constant, we can just go ahead and multiply it to each of them component -wise.

01:41 So this would be 1 over m cosine t.

01:46 One over m sine t.

01:50 All right, so we have this now for our acceleration.

01:55 So we go ahead and integrate each side of this.

02:00 So we'll end up with our velocity function here.

02:04 And then remember when we integrate a vector, we can just integrate it component -wise.

02:08 So this is going to be 1 over m integral cosine t d t.

02:15 And remember we could just pull that 1 over m out front.

02:20 And then it'd be 1 over m integral of sine t d t.

02:27 Okay.

02:29 Now we can integrate each of those, so integrating cosine, we end up with sine of of t and then i'll just say plus c1 and then over here we'd end up with negative cosine so this should be negative 1 over m sine oh sorry cosine cosine cosine plus c1 or a c2 just to distinguish it from the other one now and we can solve for c1 c2 since we know that when we plug in zero here, we should end up with just v0 .j.

03:18 So in the bracket notation, that would be 0, v .0.

03:22 Alright, so we plug in 0, so v.

03:26 0 is going to be, well, sign of 0 is 0, so that's just c1 here.

03:32 And then cosine of 0 is 1, so that should give us negative m plus c2, and then this is going to be equal to 0, v, not...

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