An object of mass 2kg fall a distance of 5m on a horizontal surface and rebounds to a vertical height of 3m. Calculate the change in momentum.
starting m g h = 2 *g * 5 Joules
so
(1/2) 2 v^2 = 10 g
v^2 = 10 g
if g = 10 m/s^2
v = -sqrt 100 = -10 m/s
momentum = -20 kgm/s
no w upward
m g h = 2 * 10 * 3 = 60
so
(1/2) 2 v^2 = 60
v = +sqrt 60
m v = 2 sqrt 60 = 15.5 kg m/s up
change = 15.5 - (-20) = 35.5 kg m/s
no change in momentum.
It had zero momentum at the start, and at the top of the bounce it is zero again.
LOL - true, but usually they mean the impulse at the turn.
To calculate the change in momentum, we need to find the initial momentum (before the collision) and the final momentum (after the collision), and then subtract the initial momentum from the final momentum.
1. Find the initial momentum (before the collision):
The initial momentum of an object can be calculated using the formula:
Initial momentum = mass × initial velocity
In this case, the object is falling vertically, so the initial velocity is the velocity just before it hits the horizontal surface. We can find this velocity using the equation of motion:
v² = u² + 2as
Where:
v = final velocity (0 m/s since it rebounds)
u = initial velocity (unknown)
a = acceleration due to gravity (-9.8 m/s²)
s = distance (5 m)
Rearranging the equation, we get:
u = √(v² - 2as)
u = √(0² - 2(-9.8)(5))
u = √(0 - (-98))
u = √(98)
u ≈ 9.9 m/s
Now, we can calculate the initial momentum:
Initial momentum = mass × initial velocity
Initial momentum = 2 kg × 9.9 m/s
Initial momentum ≈ 19.8 kg·m/s
2. Find the final momentum (after the collision):
The final momentum of the object can be calculated using the formula:
Final momentum = mass × final velocity
Since the object rebounds to a vertical height of 3 m, it gains potential energy that is equal to the loss in kinetic energy during the collision. Therefore, the final velocity can be found using the conservation of energy principle:
Initial kinetic energy = Final kinetic energy + Final potential energy
The initial kinetic energy is given by:
Initial kinetic energy = 0.5 × mass × initial velocity²
The final kinetic energy is given by:
Final kinetic energy = 0.5 × mass × final velocity²
The final potential energy is given by:
Final potential energy = mass × g × height
where g is the acceleration due to gravity (9.8 m/s²) and height is 3 m.
Rearranging the conservation of energy equation, we get:
0.5 × mass × initial velocity² = 0.5 × mass × final velocity² + mass × g × height
Substituting the values:
0.5 × 2 × 9.9² = 0.5 × 2 × final velocity² + 2 × 9.8 × 3
Solving this equation will give us the final velocity.
3. Calculate the change in momentum:
Now that we have the initial momentum and the final momentum, we can find the change in momentum by subtracting the initial momentum from the final momentum:
Change in momentum = final momentum - initial momentum
Substituting the values, we can calculate the change in momentum.