What net force would be required to accelerate an 80.0 kg man at a constant acceleration of 4.52 m/s^2, assuming no friction or other resistance is present?
We apply Newton's Second Law to solve this problem. We know that the magnitude of the force is equal to the product of the mass and the acceleration, so F = m a = 80.0 kg * 4.52 m/s^2 = 362 kgm/s^2 = 362 N.
Write the net ionic equation for the reaction of sodium carbonate and barium chloride in an aqueous solution.
Note that the chemical formula for sodium carbonate is Na2CO3, and the formula for barium chloride is BaCl2. We start off by writing out all of the ionic species initially present in the solution: 2Na[+] (aq) + CO3[2-] (aq) + Ba[2+] (aq) + 2Cl[-] (aq) -> ? This is an ion exchange reaction, so our products will be sodium chloride and barium carbonate. Referring to our solubility rules, we see that barium carbonate is insoluble in water, so the species present in the solution after the reaction has taken place will be: 2Na[+] (aq) + CO3[2-] (aq) + Ba[2+] (aq) + 2Cl[-] (aq) -> BaCO3 (s) + 2Na[+] (aq) + 2Cl[-} (aq) We see that Na[+] and Cl[-] are spectator ions - they remain unchanged by the reaction, so we can "cancel them out". What remains is the net ionic equation: Ba[2+] (aq) + CO3[2-] (aq) -> BaCO3 (s).
Find the indefinite integral of the function f(x) = 4cos(3x).
Note that we can simplify this problem using the u-substitution technique. If we substitute u = 3x, we get du = 3dx, i.e. dx = du / 3, and our expression under the integral becomes 4cos(u)du/3 = (4/3) cos(u) du. Since the integral of cos(u) du is equal to sin u, the solution is (4/3) sin u + C = (4/3) sin (3x) + C, where C is a constant.