A penny falls from a windowsill 25.0 m above the sidewalk. How fast is the penny moving when it strikes the ground? (Remember that the velocity will be negative due to the downward motion of the penny)
To calculate the speed of the penny when it strikes the ground, we can use the laws of motion and the principle of conservation of energy.
Step 1: Determine the initial potential energy of the penny when it is on the windowsill.
The potential energy (PE) of an object is given by the equation PE = mgh, where m is the mass of the object, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the height of the object above the reference point (sidewalk in this case).
Since the penny is falling freely, we can neglect any initial or final kinetic energy. Therefore, all of the initial potential energy is converted to kinetic energy when it strikes the ground.
Step 2: Calculate the velocity of the penny when it strikes the ground.
The kinetic energy (KE) of an object is given by the equation KE = (1/2)mv^2, where m is the mass of the object and v is its velocity.
Since the initial potential energy was fully converted to kinetic energy, we can equate the two equations:
mgh = (1/2)mv^2
Simplifying the equation, we get:
v^2 = 2gh
Taking the square root of both sides, we can solve for v:
v = √(2gh)
Step 3: Plug in the values and calculate.
In this case, h = 25.0 m (given in the question) and g = 9.8 m/s^2 (acceleration due to gravity):
v = √(2 * 9.8 m/s^2 * 25.0 m)
v ≈ √(490 m^2/s^2)
v ≈ 22.1 m/s
Therefore, the penny is moving at approximately 22.1 m/s (in the downward direction) when it strikes the ground.
To determine the speed at which the penny hits the ground, we can use the laws of motion and the principles of kinematics. The key equation we can use is the free-fall equation:
v^2 = u^2 + 2as
Where:
v = final velocity (in this case, the velocity of the penny just before it hits the ground)
u = initial velocity (which is 0 since the penny starts from rest on the windowsill)
a = acceleration (which is the acceleration due to gravity, approximately 9.8 m/s^2)
s = displacement (the distance the penny has fallen, which is 25.0 m)
Now we can plug in the values into the equation:
v^2 = 0 + 2 * 9.8 * 25.0
Simplifying the equation:
v^2 = 2 * 9.8 * 25.0
v^2 = 490
To find v, we take the square root of both sides:
v = √490
v ≈ 22.14 m/s
Since the velocity is negative due to the downward motion of the penny, the final speed at which the penny hits the ground is approximately 22.14 m/s, but in the negative direction.
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