if a penny is dropped from the top of a 320 ft. building, how fast will it be moving when it hits the ground?
So we can find the speed of the penny as instantaneous rate of change of height of penny when it hits ground.
Instantaneous rate of change of y with respect to x is dy/dx?
yes to
" we can find the speed of the penny as instantaneous rate of change of height of penny when it hits ground. "
Details:
If you have been given the equation
v²-u²=2aH
then
H=distance, and
velocity=v (u=constant=0)
If you express v as a function of H, which you have already done with numbers, then you get the speed of the coin (ignoring air resistance) as a function of H.
Yes, that's correct. The speed of the penny when it hits the ground can be found by determining the instantaneous rate of change of its height with respect to time, which is given by the derivative of the height function.
Yes, you are correct. In order to find the speed of the penny when it hits the ground, we need to find the instantaneous rate of change of its height.
The instantaneous rate of change of a function y with respect to the independent variable x is denoted as dy/dx. In this case, y represents the height of the penny and x represents time.
To find dy/dx for the height of the penny, we can use the laws of motion. Assuming negligible air resistance, we can use the equation of motion for an object in free fall:
y = (1/2)gt^2
where:
- y is the height of the penny (320 ft in this case)
- g is the acceleration due to gravity (32.2 ft/s^2)
- t is the time it takes for the penny to reach the ground
Differentiating both sides of the equation with respect to time (t), we get:
dy/dt = gt
Now, we need to determine the time it takes for the penny to reach the ground. We can do this by setting the height equal to zero and solving for t:
0 = (1/2)gt^2
Simplifying the equation, we get:
t^2 = 2y/g
Taking the square root of both sides:
t = sqrt(2y/g)
Substituting the given value of y (320 ft) and g (32.2 ft/s^2), we can calculate the time it takes for the penny to hit the ground.
Once we have the value of t, we can substitute it back into the equation for dy/dt:
dy/dt = gt
Substituting the values of g and t, we can calculate the instantaneous rate of change of height.
Finally, the speed of the penny when it hits the ground is equal to the magnitude of the velocity, which is given by:
v = dy/dt
After finding the value for dy/dt, we can determine the speed at which the penny hits the ground.