1)
When the person throws the ball away, the person rolls backward. This is due to the law of conservation of momentum: in fact, the total momentum of the person+ball system must be conserved.
At the beginning,
after throwing the ball, the total momentum is the sum of the momentum of the person and of the ball:
Since momentum is conserved,
So
Therefore, the person has equal momentum (in magnitude) but opposite direction to the ball, so the person rolls backward.
However, if the person hold to the ball, then they will have same momentum (moving in the same direction). In order to conserve the total momentum (which was zero at the beginning), the only possible solution is that
which means that both the person and the ball will remain at rest. This is because there are no external forces acting on the system, so the system cannot move.
2)
The change in momentum of an object is given by
where
m is the mass of the object
v is its final velocity
u is the initial velocity
For the clay ball in this problem, we have:
m = 50 g = 0.050 kg
v = 0 m/s (it sticks on the wall)
u = 1 m/s
So its change in momentum is
For the superball, we have:
m = 50 g = 0.050 kg
v = -0.8 m/s (it bounces back)
u = 1 m/s
So its change in momentum is
So, the superball has a greater change in momentum (in magnitude).
3a)
According to Newton's third law of motion:
"When an object A exerts a force (action force) on an object B, then object B exerts an equal and opposite force (reaction force) on object A".
Here, we have a Hummer and a Beetle colliding head-on: we can identify them as object A and object B. Therefore, according to Newton's third law:
- The action force is the force of impact exerted by the Hummer on the Beetle
- The reaction force is the force of impact exerted by the Beetle on the Hummer
And according to the Law, the two forces are equal in magnitude: so, the two vehicles experience the same force of impact.
3b)
The change in momentum of each vehicle during the collision can be written as
(1)
where
is the change in momentum
is the force experienced by the vehicle
is the duration of the collision
in part 3a), we said that the two vehicles experience the same force in the collision.
Moreover, the duration of the collision, , is the same for the two vehicles.
As a result, according to formula (1), the two vehicles have same change in momentum (however, the directions would be opposite, since they experience force in opposite directions).
3c)
According to Newton's second law of motion, the acceleration of an object is given by:
where
F is the force experienced by the object
m is its mass
a is its acceleration
In part 3a), we stated that the force experienced by the Beetle and the Hummer is the same. However, the mass of the Beetle is smaller than the mass of the Hummer: from the equation we see that the acceleration is inversely proportional to the mass, therefore the Beetle will experience a greater acceleration.
4a)
The force experienced by the dashboard on the car is given by:
Where
is the change in momentum
is the duration of the collision
In a padded dashboard, the duration of the collision is larger than the duration of the collision for a hard dashboard. According to the equation above, the force experienced by the dashboard (and so, the car) is inversely proportional to the duration of the collision: therefore, since the padded dashboard has a larger , it will experience a smaller force than the hard dashboard.
4b)
The force experienced by the climber if falling is given by
Where
F is the force experienced by the climber
is his change in momentum
is the duration of fall
Nylon is a very elastic material, so it is able to "soften" the fall by stretching a lot. As a result, the nylon increases the value of in the formula. Since the force experienced by the climber is inversely proportional to , the climber will feel less force thanks to the nylon.
4c)
This technique is used to exploit the "push" given by the second car of the train to the first car when the brakes are applied.
At first, the engine is started, and the first car starts accelerating, pulling the second car (and the following cars). Then, the brakes are applied on the first car: however, the second car keeps moving by inertia, so then it gives a push forward on the first car. Then, this action is repeated several times, so that this push exerted by the second car is exploited several times.
