a ball is thrown upwards from the top of a 600 foot building and reaches 664 feet before falling back down to the ground.
determine the position of and velocity function for the ball.
Determine the average velocity on the interval 1,3
Find the instantaneous velocity at 3
Find the time required for the ball to reach ground level.
Find the velocity of the ball at impact.
To determine the position and velocity functions for the ball, we can use basic principles of physics.
1. Position function:
The position function describes the height of the ball as a function of time. Let's assume the ball is thrown upwards, and the starting point is at 0 feet. We'll use the equation y = h0 + v0t - 16t^2, where y is the current height, h0 is the initial height, v0 is the initial velocity, and t is the time in seconds.
In this case, the initial height is 600 feet, the initial velocity is the upward velocity of the ball, and we'll assume it to be positive. So the position function becomes y = 600 + vt - 16t^2.
2. Velocity function:
The velocity function describes the rate at which the ball's position changes with respect to time. It is the derivative of the position function. Taking the derivative of the position function, we get dy/dt = v - 32t.
Now, we'll calculate the answers to the specific questions:
3. Average velocity on the interval [1, 3]:
To find the average velocity, we need to divide the change in position by the change in time. So, the average velocity on the interval [1, 3] is (y(3) - y(1)) / (3 - 1). Substitute the given values into the position function to find y(3) and y(1), and calculate the average velocity.
4. Instantaneous velocity at t = 3:
To find the instantaneous velocity at t = 3, we can substitute t = 3 into the velocity function dy/dt = v - 32t and solve for v.
5. Time required for the ball to reach ground level:
To find the time required for the ball to reach the ground level, we need to set y = 0 in the position function and solve for t.
6. Velocity of the ball at impact:
The velocity of the ball at impact is the velocity just before it hits the ground. To find this, substitute the time value, found in the previous step, into the velocity function to calculate the velocity at impact.
Remember to substitute the values given in the problem into the equations to get the exact numerical values for each question.