A ball is thrown upward from a height of 100 ft. with an initial velocity of 80 ft/sec. Find the function.
To find the function that describes the motion of the ball, we can use the equation of motion for objects in free fall:
s(t) = s_0 + v_0t - (1/2)gt^2
Where:
- s(t) represents the height of the ball at time t
- s_0 is the initial height (in this case, 100 ft)
- v_0 is the initial velocity (in this case, 80 ft/sec)
- g is the acceleration due to gravity (approximately 32 ft/sec^2)
Plugging in the given values, the equation becomes:
s(t) = 100 + 80t - (1/2)(32)t^2
Simplifying, we have:
s(t) = 100 + 80t - 16t^2
Therefore, the function that describes the motion of the ball is:
s(t) = 100 + 80t - 16t^2
To find the function that represents the height of the ball as a function of time, we need to consider the physics principles involved. In this case, we can use the equation of motion for an object in free fall:
h(t) = -16t^2 + V₀t + h₀,
where:
- h(t) represents the height of the object at time t,
- t is the time in seconds,
- V₀ is the initial velocity in ft/sec, and
- h₀ is the initial height in ft.
Substituting the given values into the equation, we have:
h(t) = -16t^2 + 80t + 100.
Thus, the function that represents the height of the ball as a function of time is h(t) = -16t^2 + 80t + 100.