A meteor has a speed of 90.0 m/s when 900 km above the Earth. It is falling vertically (ignore air resistance) and strikes a bed of sand in which it is brought to rest in 3.19 m.
(a) What is its speed before striking the sand?
(b) How much work does the sand do to stop the meteor (mass = 565 kg)?
(c) What is the average force exerted by the sand on the meteor?
(d) How much thermal energy is produced?
To solve this problem, we can apply the principles of conservation of energy and Newton's laws of motion.
(a) To find the meteor's speed before striking the sand, we need to use the principle of conservation of mechanical energy. At its initial position 900 km above the Earth, the meteor only has potential energy due to its height. As it falls and reaches the bed of sand, it will lose potential energy and gain kinetic energy. At the point of impact, all of its initial potential energy will have been converted into kinetic energy.
The potential energy (PE) of an object at height h is given by the formula: PE = mgh, where m is the mass, g is the acceleration due to gravity, and h is the height.
In this case, the meteor is 900 km (or 900,000 m) above the Earth, so its initial potential energy is: PE_initial = (565 kg) * (9.8 m/s^2) * (900,000 m).
The kinetic energy (KE) of an object is given by the formula: KE = 0.5mv^2, where v is the velocity.
At impact, the meteor will have lost all its potential energy and gained an equal amount of kinetic energy. So, we can equate the two expressions and solve for the velocity before striking the sand:
PE_initial = KE_final
(m * g * h) = (0.5 * m * v^2)
Canceling out the mass, we get:
g * h = 0.5 * v^2
Rearranging the equation and solving for v, we find:
v^2 = 2 * g * h
v = sqrt(2 * (9.8 m/s^2) * (900,000 m))
Calculating v will give you the speed of the meteor before striking the sand.
(b) To find the work done by the sand to stop the meteor, we can use the work-energy principle. The work done on an object is equal to its change in kinetic energy. Since the meteor comes to rest, its final kinetic energy is zero.
The work done by the sand (W) is given by the equation: W = KE_final - KE_initial
Since KE_final is zero, we can simplify the equation to: W = -KE_initial
Substituting the expression for initial kinetic energy, we get:
W = -(0.5 * m * v^2)
Plugging in the mass and the calculated velocity from part (a), you can find the work done by the sand.
(c) The average force exerted by the sand on the meteor can be calculated using Newton's second law: force = mass * acceleration.
In this case, the acceleration is equal to the change in velocity divided by the time taken to bring the meteor to rest.
Acceleration = (final velocity - initial velocity) / time
Substituting the known values, we can find the average force exerted by the sand.
(d) The amount of thermal energy produced can be calculated by subtracting the initial kinetic energy from the final kinetic energy. Since the meteor comes to rest, the final kinetic energy is zero. The thermal energy produced is then equal to the initial kinetic energy.
Thermal energy = KE_initial
Substituting the mass and the calculated velocity from part (a), you can calculate the thermal energy produced.