A particle of mass m, initially at rest at x = 0, is accelerated by a force that increases in time as F=Ct^2.
a) Determine its velocity v as a function of time.
b) Determine its position x as a function of time.
Express your answers in terms of the given quantities
Nevermind, I got it.
a) v = Ct^3/3m
b) x = Ct^4/12m
Hopefully this will be of help to someone else!
a) To determine the particle's velocity as a function of time, we need to integrate the force with respect to time. Since we know that F = ma, we can equate the force equation to m times the particle's acceleration:
Ct^2 = ma
Since the particle is initially at rest, its initial velocity v0 at t = 0 is zero. Therefore, we can integrate the right-hand side of the equation:
∫a dt = ∫(Ct^2/m) dt
Integrating gives:
a = (C/m)(t^3/3) + v0
Since the initial velocity is zero, we can simplify it to:
a = (C/m)(t^3/3)
Next, integrating the acceleration gives us the expression for velocity:
∫a dt = ∫(C/m)(t^3/3) dt
v = (C/3m)(t^4/4) + v0
Again, since the initial velocity is zero, we can simplify it to:
v = (C/3m)(t^4/4)
b) To determine the particle's position as a function of time, we integrate the velocity equation:
∫v dt = ∫[(C/3m)(t^4/4)] dt
x = (C/3m)(t^5/20) + x0
Since the particle is initially at x = 0, we can simplify it to:
x = (C/3m)(t^5/20)
So, the velocity v is given by (C/3m)(t^4/4), and the position x is given by (C/3m)(t^5/20).
To solve this problem, we can use Newton's second law of motion, which states that the acceleration of an object is equal to the net force acting on it divided by its mass:
a = F/m
In this case, the force acting on the particle is given by F = Ct^2.
a) Determine velocity v as a function of time:
To find the velocity as a function of time, we need to integrate the acceleration with respect to time:
∫ a dt = ∫ (Ct^2 / m) dt
Integrating, we obtain:
v = ∫ (Ct^2 / m) dt = (C/m) ∫ t^2 dt
The integral of t^2 is (1/3)t^3, so we have:
v = (C/m) (1/3)t^3 + A
Where A is the integration constant. Since the particle is initially at rest at x = 0, its initial velocity v₀ is zero. Therefore, A = 0, and the equation for velocity becomes:
v = (C/m) (1/3)t^3
b) Determine position x as a function of time:
To find the position as a function of time, we need to integrate the velocity with respect to time:
∫ v dt = ∫ [(C/m) (1/3)t^3] dt
Integrating, we obtain:
x = ∫ [(C/m) (1/3)t^3] dt = (C/m) (1/12)t^4 + B
Where B is the integration constant. Since the particle is initially at rest at x = 0, its initial position x₀ is also zero. Therefore, B = 0, and the equation for position becomes:
x = (C/m) (1/12)t^4
So, the expressions for velocity and position as functions of time are:
a) Velocity: v = (C/m) (1/3)t^3
b) Position: x = (C/m) (1/12)t^4
To determine the velocity and position of the particle as a function of time, we need to use Newton's second law of motion, which states that the force acting on an object is equal to its mass multiplied by its acceleration (F = ma). In this case, we have the force as a function of time, F = Ct^2, and the particle's mass as m.
a) To find the velocity v as a function of time, we first need to find the acceleration. Since F = ma, we can rearrange it to solve for a: a = F/m = (Ct^2)/m.
Next, we can use the definition of acceleration, which is the derivative of velocity with respect to time: a = dv/dt. We can solve this differential equation by integrating both sides with respect to time:
∫a dt = ∫(Ct^2)/m dt
Integrating the left side gives us v, the velocity, and integrating the right side gives us the antiderivative of (Ct^2)/m:
v = ∫(Ct^2)/m dt
Integrating (Ct^2)/m with respect to t will give us (C/3m)t^3 plus a constant of integration. Since the particle is initially at rest, we can set this constant to zero. Therefore, the expression for velocity as a function of time is:
v = (C/3m)t^3
b) To find the position x as a function of time, we need to integrate the velocity function we found in part a with respect to time:
x = ∫(C/3m)t^3 dt
Integrating (C/3m)t^3 with respect to t will give us (C/12m)t^4 plus another constant of integration. Setting this constant to zero, since the particle is initially at x = 0, gives us the expression for position as a function of time:
x = (C/12m)t^4
So, the velocity v as a function of time is given by v = (C/3m)t^3, and the position x as a function of time is given by x = (C/12m)t^4.