A particle of mass m is subjected to a net force 𝐅(t) given by 𝐅(t)=F0(1-t / T) 𝐢̂; that is, F(t

A particle of mass m is subjected to a net force 𝐅(t) given...

A particle of mass $m$ is subjected to a net force $\mathbf{F}(t)$ given by $\overrightarrow{\mathbf{F}}(t)=F_{0}(1-t / T) \hat{\mathbf{i}}$; that is, $F(t)$ equals $F_{0}$ at $t=0$ and decreases linearly to zero in time $T .$ The particle passes the ori$\operatorname{gin} x=0$ with velocity $v_{0} \hat{\mathbf{i}}$. Show that at the instant $t=T$ that $F(t)$ vanishes, the speed $v$ and distance $x$ traveled are given by $v(T)=v_{0}+a_{0} T / 2, \quad$ and $x(T)=v_{0} T+a_{0} T^{2} / 3$, where $a_{0}=F_{0} / m$ is the initial acceleration. Compare these results with Eqs. $2-26$ and $2-28$.

A particle of mass $m$ is subjected to a net force $\mathbf{F}(t)$ given by $\overrightarrow{\mathbf{F}}(t)=F_{0}(1-t / T) \hat{\mathbf{i}}$; that is, $F(t)$ equals $F_{0}$ at $t=0$ and decreases linearly to zero in time $T .$ The particle passes the ori$\operatorname{gin} x=0$ with velocity $v_{0} \hat{\mathbf{i}}$. Show that at the instant $t=T$ that $F(t)$ vanishes, the speed $v$ and distance $x$ traveled are given by $v(T)=v_{0}+a_{0} T / 2, \quad$ and $x(T)=v_{0} T+a_{0} T^{2} / 3$,
where $a_{0}=F_{0} / m$ is the initial acceleration. Compare these results with Eqs. $2-26$ and $2-28$.

Key Concepts

Integration of Velocity to Determine Displacement

Similarly, to find the displacement or position of the particle, the velocity function must be integrated with respect to time. This definite integration, often performed after finding the velocity function, involves applying initial position conditions to yield an expression for the displacement that accounts for the variable acceleration throughout the period under consideration.

Integration of Acceleration to Determine Velocity

With a known time-dependent acceleration, velocity is found by integrating the acceleration function with respect to time. This process involves computing the definite integral of the acceleration function over the relevant time interval and applying the initial conditions to obtain a complete expression for velocity as a function of time.

Time-Dependent (Variable) Acceleration

When the applied force is not constant but depends on time, the resulting acceleration is also a function of time. This scenario requires the use of calculus to handle the integration of the acceleration function. Variable acceleration complicates the direct use of the standard kinematic equations that assume constant acceleration and requires tailored integration methods to derive velocity and displacement.

Newton's Second Law

Newton's Second Law establishes the relationship between force and acceleration, stating that the net force acting on a particle is equal to the mass of the particle multiplied by its acceleration (F = ma). In contexts with time-varying forces, this law serves as the starting point for determining how the particle's acceleration changes over time and ultimately how its velocity and displacement evolve.

Transcript

00:01 We are going to prove two expressions and we have been given with the expression for force, force vector.

00:08 So here you can see the force vector is given like this.

00:11 Ft vector is f0 times 1 minus small t by capital t i can.

00:19 And it acts on a particle, right? now here this particular force tells us that at zero time the force should have been f0, right? and at t time the force will eventually decrease to zero as it is given in the question now we have to prove first of all the expressionable velocity at the time t will be given by v0 plus a not t by two right and x t similarly will be given by v0 t plus a naught t square by three these two we have to derive and we have an information with us that a naught is f0 by m right so the initial force divided by the mass of the particle.

01:03 Now, in order to find out the velocity, what we are going to do, we are going to use the newton's second law, right? so first of all, let us write newton's second law.

01:17 And newton's second law basically tells you that any force vector is given by mass of the particle times its acceleration vector.

01:28 So here we will be taking it to our further step and we'll be writing the acceleration vector as the velocity vector divided by the time.

01:37 So the differential of velocity is given by acceleration.

01:43 Now we'll be substituting the values for ft here.

01:46 So ft is nothing but f0 1 minus t by t.

01:50 This is the expression for ft and here we are just writing m.

01:54 Dv by d t let's say so now we can do one thing we can take this m to the left and we can take this d t to the left as well so we'll be getting something like this uh an expression something like this f0 divided by m times one minus t by t right d t and that will be equal to our dv and then we can simply integrate right with to t for the left side and with respect to v on the right side.

02:31 So i am putting the integration symbol here and we are doing the integration for the time 0 to t capital t and we are doing the integration for velocity starting from let's say v some value v and ending with our expression the velocity that we are requiring here vt right...

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