A car of mass 1880 kg is speeding up on level ground. Its position as a function of time is
x(t)=Att1+t41t4
where A=527 m and t1=51 s.
What is the magnitude of the net impulse on the car from t=0 to t= 0.5 s?
To find the magnitude of the net impulse on the car from t=0 to t=0.5 s, we need to calculate the change in momentum of the car during this time interval.
The change in momentum Δp is equal to the impulse J applied to an object. The impulse is given by the equation:
J = ∫F dt
Where F is the net force acting on the car and t is the time interval.
In this case, we have the position function x(t) = Att^1 + t^4, which allows us to calculate the velocity and acceleration of the car. The velocity v(t) is the derivative of the position function with respect to time:
v(t) = d/dt (Att^1 + t^4)
v(t) = 2At + 4t^3
The acceleration a(t) is the derivative of the velocity function with respect to time:
a(t) = d/dt (2At + 4t^3)
a(t) = 2A + 12t^2
To find the net force acting on the car, we multiply the mass of the car (m = 1880 kg) by the acceleration:
F = m * a(t)
F = 1880 * (2A + 12t^2)
Now, we can integrate the force function F with respect to time from t = 0 to t = 0.5 s to find the impulse J:
J = ∫F dt
J = ∫(1880 * (2A + 12t^2)) dt
Integrating the force equation with respect to time gives us:
J = ∫(3760A + 22560t^2) dt
J = [3760At + 7520t^3] from 0 to 0.5
Calculating the impulse J by substituting t = 0.5 and t = 0 into the equation:
J = (3760A * 0.5 + 7520 * 0.5^3) - (3760A * 0 + 7520 * 0^3)
J = (1880A + 470) - 0
J = 1880A + 470
Finally, substituting the given values A = 527:
J = 1880(527) + 470
J = 989,960 + 470
J = 990,430 Ns
Therefore, the magnitude of the net impulse on the car from t = 0 to t = 0.5 s is 990,430 Ns.