Tutor profile: Casey D.
Subject: Physics (Newtonian Mechanics)
A ball is kicked from ground level at an angle of 30 degrees above the ground with an initial velocity of 10 m/s. How far from its initial position does the ball land and what is its maximum height?
So to find the balls maximum height, we need to find the point where it stops rising, is completely still for a split second, before it starts falling. This means that the vertical velocity will be zero. Using the kinematic equation v_final^2 = v_initial^2 + 2*a*d and plugging in our values (rounding the acceleration of gravity to -9.8) we can get: 0 = (10*sin(30))^2 + 2(-9.8)(d) Solving gives d = 1.275 m for our maximum height Before moving onto the horizontal distance traveled we have to figure out the time it took the ball to reach that height, then double it to get the total time spent in the air. Using the kinematic equation v_final = v_initial + a*t and plugging in our values: 0 = (10*sin(30)) + (-9.8)(t) t = 0.51 seconds, which doubled gives us 1.02 seconds for total flight time Now, since the ball has no horizontal acceleration (because after the initial kick there are no more horizontal forces acting on it which means the horizontal velocity stays constant) we can use the common equation d = v_initial*t and plug in our values: d = (10*cos(30))*1.02 Solving gives d = 8.83 meters Therefore, our final answers are the ball reaches a maximum height of 1.275 meters and lands 8.83 meters away from its initial location.
What is the inverse of the function f(x) = 4x^2 - 3 ?
To find the inverse of a basic function such as the one shown above, all we have to do is switch the variables: Re-writing the function as y = 4x^2 - 3 to replace f(x) with y We can now switch the x and y in the equation to get: x = 4y^2 - 3 The final step would be to isolate the y so the equation is back to being a function of x: x = 4y^2 - 3 x+3 = 4y^2 (x+3)/4 = y^2 y = (1/2)*sqrt(x+3)
Evaluate f(g(3)) for f(x) = (2x - 1)/4 and g(x) = 4(x^2) - 4:
When working with nested functions such as the one shown above we always work from inside to outside. Essentially, this means that we take the g(x) function and plug it in where the x is in the f(x) function. However, since we are given a value for our x variable, the problem can be made more simple by plugging the 3 into g(x) before we integrate the functions. g(3) = 4*(3^2) - 4 = 4*9 - 4 = 36 - 4 = 32 f(g(3)) = (2*32 - 1)/4 = (61 - 1)/4 = 63/4 So our final answer is 63/4 or 15.75
needs and Casey will reply soon.