you throw a wad of used paper towards a wastebasket from a height of about 1.3 feet above the floor with an initial vertical velocity of 3 feet per second.
a. write and graph a function that models the height h (in feet) of the paper t seconds after it is thrown
b. if you miss the wastebasket and the paper hits the floor, how long does it take for the ball of paper to reach the floor.
c. if the ball of paper hits the rim of the wastebasket one-half foot above the ground, how long was the ball in the air?
a. To model the height h (in feet) of the paper t seconds after it is thrown, we need to consider the initial height, initial velocity, and the effect of gravity.
The equation for the height of an object in free-fall motion (ignoring air resistance) is given by:
h(t) = -16t^2 + v0t + h0
where:
h(t) is the height of the object at time t
t is the time in seconds
v0 is the initial vertical velocity (in feet per second)
h0 is the initial height (in feet)
In this case, the initial vertical velocity is 3 feet per second and the initial height is 1.3 feet above the floor. Therefore, the function that models the height of the paper is:
h(t) = -16t^2 + 3t + 1.3
To graph this function, plot time (t) on the x-axis and height (h) on the y-axis, using suitable units. The graph will be a parabola opening downwards.
b. To find how long it takes for the paper to reach the floor, we need to determine when the height is zero (h = 0). We can solve this by setting the equation equal to zero and solving for t:
-16t^2 + 3t + 1.3 = 0
Using the quadratic formula, t = (-b ± √(b^2 - 4ac)) / (2a), where a = -16, b = 3, and c = 1.3, we can find the value of t. Since we want to find the positive solution, we ignore the negative value obtained from the quadratic formula.
c. To calculate how long the ball of paper was in the air, we need to determine when it reaches the rim of the wastebasket. In this case, the height is 0.5 feet above the ground (h = 0.5). We can again use the quadratic formula to solve for t:
-16t^2 + 3t + 1.3 = 0.5
By setting the equation equal to 0.5 and solving for t, we can determine the time it takes for the ball of paper to hit the rim of the wastebasket.
a. To model the height h (in feet) of the paper at time t (in seconds), we can use the formula for the vertical motion under constant acceleration:
h(t) = -16t^2 + vt + h0
Where:
- t is the time in seconds,
- v is the initial vertical velocity (3 feet per second),
- h0 is the initial height (1.3 feet) above the floor.
In this case, the initial vertical velocity is upward, so we consider it as positive. However, the formula uses gravity (-16 ft/s^2) with a negative sign.
Therefore, the function that models the height h(t) is:
h(t) = -16t^2 + 3t + 1.3
To graph this function, plot points where you substitute values of t into the equation to find the corresponding heights h:
For example:
- When t = 0, h(0) = -16(0)^2 + 3(0) + 1.3 = 1.3, so the initial height is 1.3 ft.
- When t = 1, h(1) = -16(1)^2 + 3(1) + 1.3 = -11.7, so the height is -11.7 ft after 1 second.
You can continue to calculate more points, and then plot them on a graph with time (t) on the x-axis and height (h) on the y-axis.
b. To find how long it takes for the paper to hit the floor, we need to solve for t when h(t) = 0. Since the height h represents the vertical position, when the paper hits the floor, h = 0.
So, we set the equation -16t^2 + 3t + 1.3 = 0 and solve for t. This can be done by factoring the equation or using the quadratic formula.
c. If the ball of paper hits the rim of the wastebasket one-half foot above the ground, we set h(t) = 0.5 and solve for t. This will give us the amount of time the ball was in the air.