Final answer:
To determine the object's speed at t=3.0s with a mass of 10kg and starting velocity of 0.0m/s, based on the force Fx(t) = (5 + 3t²)N, we calculate the acceleration by dividing the force by the mass and integrate this function from t=0 to t=3s to obtain the velocity, resulting in a speed of 4.2 m/s.
Step-by-step explanation:
To calculate the object's speed at t=3.0s based on the given force equation Fx(t) = (5 + 3t²)N, we must integrate the force over time to find the change in momentum, which is equal to the change in velocity when mass is constant. Since the mass of the object is 10kg and the initial velocity is 0.0m/s when t=0.0s, we can calculate the acceleration at any time t by using Newton's second law of motion, which states that Force equals mass times acceleration (F=ma). Therefore, the acceleration a(t) is given by Fx(t)/m. To find the final velocity v(t), we must integrate the acceleration with respect to time.
First, calculate acceleration:
- a(t) = Fx(t)/m = (5 + 3t²)/10
Then, integrate a(t) from t=0 to t=3s to find velocity:
- v(t) = ∫ a(t) dt = ∫ (5/10 + 3t²/10) dt = (1/2)t + (1/10)t³ + C
Since the initial velocity is zero, the constant of integration C is zero. Plugging t=3s into the velocity equation:
- v(3s) = (1/2)(3) + (1/10)(3)³ = 1.5 + 2.7 = 4.2 m/s
Therefore, the object's speed at t=3.0s is 4.2 m/s.