The position of a particle moving on a smooth horizontal floor is given as a function of time according to the equation x=ut+1/2at^2. At time t=0, the particle has the velocity u. Obtain an expression for the velocity as the function of time. Hence, find the value of this velocity after 8 seconds given that the initial velocity ,u, is 5.0m/s.
v(t) = dx/dt = u+at
Now go for it. I hope you have a value for a.
I want the layout of the solution please
To find the expression for the velocity as a function of time, we differentiate the equation x = ut + (1/2)at² with respect to time.
Differentiating x = ut + (1/2)at² with respect to time (t), we get:
dx/dt = d/du(ut) + d/dt((1/2)at²)
dx/dt = u + (1/2)(2at)
dx/dt = u + at
So, the expression for the velocity as a function of time is v(t) = u + at.
Given that the initial velocity, u, is 5.0 m/s, we can substitute this value into the velocity equation.
v(t) = 5.0 + at
To find the value of the velocity after 8 seconds (t = 8), we substitute t = 8 into the equation:
v(8) = 5.0 + a(8)
Since we don't have a value for acceleration (a), we can't determine the exact value of the velocity after 8 seconds without additional information.
To obtain an expression for the velocity as a function of time, we need to differentiate the position equation with respect to time.
Given: x = ut + (1/2)at^2
Differentiating both sides of the equation with respect to time:
dx/dt = d(ut)/dt + d(1/2)at^2/dt
The first term, d(ut)/dt, is simply u because the derivative of ut with respect to t is u.
For the second term, we need to apply the power rule for differentiation: d(at^2)/dt = a * d(t^2)/dt.
Using the power rule, we have:
d(t^2)/dt = 2t.
Evaluating the derivative of (1/2)at^2 with respect to t:
d(1/2)at^2/dt = (1/2) * 2 * a * t = a * t.
Combining all the terms:
dx/dt = u + a * t.
Therefore, the expression for velocity as a function of time is v = dx/dt = u + a * t.
Given that the initial velocity u is 5.0 m/s and we want to find the velocity after 8 seconds, we substitute the values into the expression:
v = u + a * t
v = 5.0 m/s + a * 8 s.
However, we don't have the value of acceleration a. To find the value of velocity after 8 seconds, we need additional information or an equation that relates velocity, time, and acceleration.