A spherical 0.34 kg orange, 2.0 cm in radius, is dropped from the top of a building of height 31 m. After striking the pavement, the shape of the orange is a 0.60 cm thick pancake. Neglect air resistance and assume that the collision is completely inelastic.(a) Estimate how much time the orange took to completely "squish" to a stop? (b) What average force did the pavement exert on the orange during the collision?
To estimate the time it took for the orange to completely "squish" to a stop, we can use the concept of conservation of mechanical energy. The initial potential energy when the orange is dropped is converted into kinetic energy as it falls, and then into elastic potential energy when it is compressed, and finally into thermal energy due to the inelastic collision.
(a) Let's break down the problem into different stages:
1. Free-fall stage: The orange falls freely for a distance equal to the height of the building. We can calculate the time it takes to fall using the equation:
Δh = (1/2)gt^2
where Δh is the height, g is the acceleration due to gravity (approximately 9.8 m/s^2), and t is the time. Rearranging the equation to solve for t, we get:
t = √(2Δh / g)
Plugging in the values, we have:
t = √(2 * 31 m / 9.8 m/s^2)
t ≈ 2.5 s
2. Collision stage: After the fall, the orange collides with the pavement and undergoes deformation. To estimate the time for complete deformation, we assume that this stage is similar to a damped harmonic motion and use an empirical relationship between the damping time and the nature of collision. For a relatively soft collision like this one, the damping time can be approximated as approximately 3 times the time taken for the object to accelerate due to gravity. So, in our case:
t_damping ≈ 3 * t ≈ 3 * 2.5 s
t_damping ≈ 7.5 s
Therefore, the estimated time it took for the orange to completely "squish" to a stop is approximately 7.5 seconds.
(b) To calculate the average force exerted by the pavement on the orange during the collision, we need to consider the change in momentum of the orange.
The initial momentum of the orange before the collision is given by:
p_initial = m * v
Where m is the mass of the orange and v is its initial velocity just before it hits the pavement.
The final momentum of the orange is zero since it comes to rest after the collision.
According to the law of conservation of momentum, we can write:
p_initial = p_final
m * v = 0
From this, we can solve for the initial velocity of the orange before the collision:
v = 0 / m
Since the collision is completely inelastic, the final velocity of the orange will be zero. The change in momentum is then given by:
Δp = p_final - p_initial
Substituting the values, we get:
Δp = 0 - (m * v)
We know that the force is defined as the rate of change of momentum, so the average force exerted by the pavement can be calculated using the formula:
F = Δp / t_damping
Substituting the values, we have:
F = -(m * v) / t_damping
Plugging in the values we have:
F = -(0.34 kg * 0 m/s) / 7.5 s
F = 0 N
Therefore, the average force exerted by the pavement on the orange during the collision is approximately 0 Newtons.