if a ball is thrown vertically upward with a velocity of 80m/s then its height after t seconds is s=80t-16t^2
a) what is the maximum height reached by the ball?
b)what is the velocity of the ball when it is 96 ft above the ground on its way up? on its way down?
please ignore, i got it by myself
6 seconds
a) To find the maximum height reached by the ball, we need to determine the turning point of the given equation.
The equation for the height of the ball is given as: s = 80t - 16t^2
To find the maximum height, we need to find the vertex of the parabolic equation. The vertex of a parabola given by the equation y = ax^2 + bx + c is given by the formula: x = -b / (2a)
In this case, the equation for the height is s = -16t^2 + 80t, so the values of a, b, and c are -16, 80, and 0 respectively.
Using the formula, we can find the value of t at the maximum height:
t = -80 / (2*(-16))
t = -80 / (-32)
t = 2.5 seconds
Now, we can substitute the value of t back into the original equation to find the maximum height:
s = 80t - 16t^2
s = 80*(2.5) - 16*(2.5)^2
s = 200 - 16*(6.25)
s = 200 - 100
s = 100 meters
Therefore, the maximum height reached by the ball is 100 meters.
b) To find the velocity of the ball when it is 96 ft above the ground on its way up and down, we first need to convert the height from feet to meters.
1 foot = 0.3048 meters
So, 96 ft * 0.3048 = 29.26 meters
On its way up, the velocity can be found by taking the derivative of the equation for height with respect to time:
v = ds/dt
v = d(80t - 16t^2)/dt
v = 80 - 32t
Now, substitute t = 2.5 seconds (time at which the ball is 96 ft above the ground) into the equation to find the velocity on its way up:
v = 80 - 32*(2.5)
v = 80 - 80
v = 0 m/s
The velocity on its way up is 0 m/s.
On its way down, the velocity can also be found by using the equation v = 80 - 32t. Since we know that the maximum height is reached at t = 2.5 seconds, we can find the velocity at any time after that.
Substitute t > 2.5 seconds into the equation to find the velocity on its way down:
v = 80 - 32t
For example, let's find the velocity when t = 3 seconds:
v = 80 - 32*(3)
v = 80 - 96
v = -16 m/s
So, the velocity on its way down when the ball is 96 ft above the ground is -16 m/s.