a ball is dropped from rest at 25m above the ground. how fast is it moving at 10 m above the ground and b) how much time does it take to reach the ground? no air resistance.
at ten meters it has dropped 15 meters
(1/2) m v^2 = m g h
v^2 = 2 * 9.81 * 15 = 294.3
v = 17.2 m/s at 10 m above ground
d = (1/2) g t^2
25 = 4.9 t^2
t = 2.26 seconds
To find the speed of the ball when it is at a height of 10 meters above the ground, we can use the principle of conservation of energy. When the ball is dropped from rest, it only has gravitational potential energy, which is converted to kinetic energy as it falls.
The gravitational potential energy (PE) of an object is given by the equation:
PE = m * g * h
where m is the mass of the object, g is the acceleration due to gravity (approximately 9.8 m/s^2 on Earth), and h is the height.
In this case, the ball has a height of 25 meters initially. Therefore, its potential energy at that height is:
PE_initial = m * g * h_initial = m * g * 25
When the ball reaches a height of 10 meters, its potential energy becomes:
PE_final = m * g * h_final = m * g * 10
Since energy is conserved, we can equate the two expressions:
PE_initial = PE_final
m * g * 25 = m * g * 10
The mass (m) cancels out, and we are left with:
25g = 10g
Now, we can solve for g:
g = g
Therefore, the acceleration due to gravity does not affect the calculation. The speed of the ball at 10 meters above the ground is the same as the speed it has when dropped from rest.
The formula to calculate the speed of an object in free fall is:
v = sqrt(2 * g * h)
where v is the speed, g is the acceleration due to gravity, and h is the height.
In this case, we can use the formula with the given values:
v = sqrt(2 * 9.8 * 10) = sqrt(196) = 14 m/s
Therefore, the ball is moving at a speed of 14 m/s when it is 10 meters above the ground.
To find the time it takes for the ball to reach the ground, we can use the formula for free fall:
h = (1/2) * g * t^2
where h is the height, g is the acceleration due to gravity, and t is the time.
In this case, the initial height is 25 meters, and we want to find the time it takes for the ball to reach the ground (h = 0).
0 = (1/2) * 9.8 * t^2
Simplifying the equation:
0 = 4.9 * t^2
Dividing both sides by 4.9:
0 = t^2
Since t^2 = 0, the time it takes for the ball to reach the ground is 0 seconds.
In conclusion, the ball is moving at a speed of 14 m/s when it is 10 meters above the ground, and it takes 0 seconds to reach the ground.