a stone thrown upward from the top of a 320-foot cliff at 128 ft/sec eventually falls to the beach below?
how long does the stone take to reach its highest point?
the height h is
h(t) = 320 + 128t - 16t^2
The vertex of this parabola is at t = 4
So, figure h(4)
To determine how long the stone takes to reach its highest point, we can use the equation of motion for vertical motion, which is given by:
h(t) = h0 + v0t - 16t^2
where:
h(t) is the height of the stone at time t
h0 is the initial height (320 feet)
v0 is the initial vertical velocity (128 ft/sec)
t is the time
At the highest point, the vertical velocity becomes zero. Therefore, we can set v0 - 16t = 0 and solve for t.
v0 - 16t = 0
128 - 16t = 0
16t = 128
t = 128/16
t = 8
Therefore, the stone takes 8 seconds to reach its highest point.
To find out how long the stone takes to reach its highest point, we can determine the time it takes for the stone to reach its highest point by using the formula for time:
Time = (Final Velocity - Initial Velocity) / Acceleration
In this case, the stone is being thrown upwards, so its velocity is decreasing due to gravity. The acceleration is the acceleration due to gravity, which is approximately -32 ft/s² (negative because the stone is moving against gravity).
Given:
Initial Velocity (U) = 128 ft/sec
Acceleration (a) = -32 ft/s²
To find the time taken to reach the highest point, we need to find the final velocity (V) when the stone reaches the highest point. At this point, the final velocity becomes zero because the stone momentarily stops before falling back down.
Using the formula for final velocity:
Final Velocity (V) = Initial Velocity (U) + (Acceleration * Time)
Setting V = 0, we can solve for Time:
0 = 128 - (32 * Time)
32 * Time = 128
Time = 128 / 32
Time = 4 seconds
Therefore, the stone takes 4 seconds to reach its highest point.