A Christmas tree ball will break if dropped on a hardwood floor with a speed of 2.0 m/s or more. If it falls 15 cm from a tree branch to the floor, will it break?
To determine whether the Christmas tree ball will break when it falls from a height of 15 cm, we need to find its final velocity just before hitting the hardwood floor.
We can use the equation of motion: vf^2 = vi^2 + 2ad, where:
- vf is the final velocity
- vi is the initial velocity (0 m/s in this case because the ball is initially at rest)
- a is the acceleration due to gravity (approximately 9.8 m/s^2)
- d is the distance fallen (15 cm = 0.15 m)
Plugging in the values, the equation becomes:
vf^2 = 0^2 + 2 * (-9.8) * 0.15
Simplifying further:
vf^2 = -2.94
Since the final velocity squared is negative, it means that the ball did not have enough speed to break upon impact. Therefore, the Christmas tree ball will not break when falling from a height of 15 cm onto the hardwood floor.
To determine if the Christmas tree ball will break when it falls from a distance of 15 cm (0.15 m) onto a hardwood floor, we need to calculate its final velocity when it reaches the floor.
We can use the principle of conservation of energy to solve this problem. The initial potential energy of the ball when it is on the tree branch will be equal to the final kinetic energy of the ball when it hits the floor.
The potential energy of an object can be calculated using the formula: PE = m * g * h
Where:
- PE is the potential energy
- m is the mass of the object
- g is the acceleration due to gravity (approximately 9.8 m/s²)
- h is the height or distance from the tree branch to the floor
Since no mass is provided, we'll assume the mass of the Christmas tree ball doesn't affect the breaking threshold.
Therefore, the initial potential energy of the ball is: PE_initial = m * g * h = 0.15 * 9.8 * 0.15
Now, let's find the equivalent kinetic energy of the ball when it hits the floor. The kinetic energy of an object can be calculated using the formula: KE = (1/2) * m * v²
Where:
- KE is the kinetic energy
- v is the velocity of the object
We need to find the final velocity (v) when the ball hits the floor with a height of 15 cm. Since the ball is dropped (no initial velocity), the initial kinetic energy is zero.
So, KE_initial = 0.5 * m * 0 = 0
Since the principle of conservation of energy states that the initial potential energy is equal to the final kinetic energy, we have:
PE_initial = KE_final
0.15 * 9.8 * 0.15 = 0.5 * m * v_final²
Simplifying the equation, we find:
v_final² = (0.15 * 9.8 * 0.15) / 0.5
v_final² = 0.02205
Finally, let's take the square root of v_final² to find the final velocity (v_final):
v_final = √0.02205
v_final ≈ 0.1487 m/s
The final velocity of the ball when it hits the floor is approximately 0.1487 m/s.
Since it is less than 2.0 m/s, the ball will not break when it falls from a height of 15 cm onto the hardwood floor.
KE at the floor: 1/2 mv^2
PE at drop = mgh
set them equal
mg(.15)=? 1/2 m (2^2)
.15g =? 2
g=? 2/.15=13
but g is not = or > than 13, so
the ball does notbreak, not enough height.