A 4.0 kg particle is moving along the x axis to the left with a velocity of v= -12.0 m/s. Suddenly, between times t =0 and t = 4.0 x a net force = 3t^2 – 12t is applied to the particle, where F is in N and t is in s. Calculate the velocity of the particle at t=4.0 s. Can I just substitute 4 in this formula 3t^2 - 12t to solve
you have to calculate acceleration.
a= force/mass= (3t^2-12t)/4
v(4)=-12+a*t
now put t=4 into the equation
To find the velocity of the particle at t = 4.0 s, you cannot just substitute 4 into the formula 3t^2 - 12t. This is because the force applied to the particle changes over time with the given equation.
To calculate the velocity, you need to use Newton's second law of motion, which relates the net force applied to an object to its acceleration:
F = ma
In this case, the mass (m) of the particle is given as 4.0 kg. The force (F) applied to the particle is described by the equation 3t^2 - 12t, where t is the time. Since the force varies with time, we need to find the resulting acceleration (a) at t = 4.0 s.
To do this, differentiate the force function with respect to time (t) to find the acceleration function (a(t)):
a(t) = d/dt (3t^2 - 12t)
Differentiating the force equation, we get:
a(t) = 6t - 12
Now, we can substitute t = 4.0 s into the acceleration equation to find the acceleration at t = 4.0 s:
a(4.0) = 6(4.0) - 12
a(4.0) = 24 - 12
a(4.0) = 12 m/s^2
Now that we have the acceleration, we can use it to find the velocity at t = 4.0 s. We can do this by integrating the acceleration function with respect to time:
v(t) = ∫ a(t) dt
v(t) = ∫ (6t - 12) dt
Integrating the acceleration equation, we get:
v(t) = 3t^2 - 12t + C
Where C is the constant of integration.
To find the constant C, we can use the initial conditions given in the problem. At t = 0, the particle has a velocity of -12.0 m/s. Substituting these values into the velocity equation:
-12 = 3(0)^2 - 12(0) + C
-12 = C
So, now we have the complete velocity equation:
v(t) = 3t^2 - 12t - 12
To find the velocity at t = 4.0 s, substitute t = 4.0 into the velocity equation:
v(4.0) = 3(4.0)^2 - 12(4.0) - 12
v(4.0) = 3(16.0) - 48.0 - 12
v(4.0) = 48.0 - 48.0 - 12
v(4.0) = -12.0 m/s
Therefore, the velocity of the particle at t = 4.0 s is -12.0 m/s, indicating that the particle is still moving to the left at the same speed.