You are standing on a sheet of ice that covers the football stadium parking lot in Buffalo; there is negligible friction between your feet and the ice. A friend throws you a 0.405 kg ball that is traveling horizontally at 11.5 m/s . Your mass is 71.5 kg .
If you catch the ball, with what speed do you and the ball move afterwards? If the ball hits you and bounces off your chest, so that afterwards it is moving horizontally at 7.80 m/s in the opposite direction, what is your speed after the collision?
Case A:
Taking your mass as M
Mass of the ball as m
Initial ball speed as u
Final combined speed as v:
Using conservation of momentum -
Initial momentum = Final momentum
mu = (M + m)v
0.405*11.5 = (71.5+0.405)V
V = 4.6575/71.905
= 0.0647 m/sec
= 6.47 cm/s
M1 = 0.405 kg, M2 = 71.5 kg.
M1*V1 + M2*V2 = M1*V + M2*V.
0.405*11.5 + 71.5*0 = 0.405V + 71.5V.
4.66 + 0 = 71.905V,
V = 0.0467 m/s. = 4.67 cm/s.
v
8
To solve this problem, we can use the principle of conservation of momentum. The total momentum before the collision is equal to the total momentum after the collision, assuming no external forces act on the system.
Let's calculate the two scenarios separately.
1. Catching the ball:
Before the ball is thrown to you, the total momentum of the system is zero, since you and the ball are not moving.
After you catch the ball, the total momentum of the system will still be zero, since there are no external forces acting on the system. This means that the momentum gained by the ball will be offset by the momentum gained by you.
Let's use the conservation of momentum equation:
(mass of ball x velocity of ball before) + (mass of you x velocity of you before) = (mass of ball x velocity of ball after) + (mass of you x velocity of you after)
(0.405 kg x 11.5 m/s) + (71.5 kg x 0 m/s) = (0.405 kg x velocity of ball after) + (71.5 kg x velocity of you after)
(0.405 kg x 11.5 m/s) = (0.405 kg x velocity of ball after) + (71.5 kg x velocity of you after)
4.6575 kg·m/s = (0.405 kg x velocity of ball after) + (71.5 kg x velocity of you after)
Since the velocity of you before the collision is 0 m/s, the equation simplifies to:
4.6575 kg·m/s = (0.405 kg x velocity of ball after) + (0 kg x velocity of you after)
4.6575 kg·m/s = (0.405 kg x velocity of ball after)
Now we solve for the velocity of the ball after:
velocity of ball after = 4.6575 kg·m/s / 0.405 kg
velocity of ball after = 11.5 m/s
Therefore, after catching the ball, both you and the ball will have a velocity of 11.5 m/s in the horizontal direction.
2. Ball bounces off your chest:
In this scenario, the momentum is not conserved as the ball bounces off your chest. Only the horizontal component of momentum is conserved.
Using the same equation as before, considering only the horizontal component of momentum:
(mass of ball x velocity of ball before) = (mass of ball x velocity of ball after) + (mass of you x velocity of you after)
(0.405 kg x 11.5 m/s) = (0.405 kg x -7.8 m/s) + (71.5 kg x velocity of you after)
4.6575 kg·m/s = (-2.997 kg·m/s) + (71.5 kg x velocity of you after)
Now we solve for the velocity of you after:
71.5 kg x velocity of you after = 4.6575 kg·m/s - (-2.997 kg·m/s)
71.5 kg x velocity of you after = 7.6545 kg·m/s
velocity of you after = 7.6545 kg·m/s / 71.5 kg
velocity of you after ≈ 0.107 m/s
Therefore, after the collision, when the ball bounces off your chest, your speed will be approximately 0.107 m/s in the opposite direction of the ball.
To solve this problem, we can apply the principle of conservation of momentum.
1. Catching the ball:
When you catch the ball, you and the ball will move together as a system. Since there are no external horizontal forces acting on the system, the total momentum before and after catching the ball will be the same.
The total momentum before catching the ball can be calculated as:
Total momentum before = Your momentum before + Ball's momentum before
Your momentum before catching the ball = mass × velocity (since there is negligible friction and you are stationary, your initial velocity is zero)
Your momentum before = 71.5 kg × 0 m/s = 0 kg·m/s
Ball's momentum before catching the ball = mass × velocity
Ball's momentum before = 0.405 kg × 11.5 m/s = 4.6575 kg·m/s
Total momentum before = 0 kg·m/s + 4.6575 kg·m/s = 4.6575 kg·m/s
After catching the ball, since you and the ball will move together, you will share the same velocity. Let's say your final velocity after catching the ball is v.
Therefore, the total momentum after catching the ball can be expressed as:
Total momentum after = (Your mass + Ball's mass) × Your velocity after catching the ball
Since your mass and the ball's mass remain the same, the total momentum will be:
Total momentum after = (71.5 kg + 0.405 kg) × v
According to the principle of conservation of momentum:
Total momentum before = Total momentum after
So, we can set up an equation:
4.6575 kg·m/s = (71.5 kg + 0.405 kg) × v
Now we can solve for v:
v = 4.6575 kg·m/s / (71.905 kg)
v ≈ 0.0648 m/s
Therefore, after catching the ball, you and the ball will move together with a velocity of approximately 0.0648 m/s.
2. Ball bouncing off your chest:
In this scenario, the ball bounces off your chest in the opposite direction. Again, we can apply the principle of conservation of momentum to solve this.
The total momentum before the collision is the same as in the previous case, which is 4.6575 kg·m/s.
After the collision, let's say your final velocity is v' in the opposite direction of the ball's velocity (7.80 m/s).
The total momentum after the collision can be expressed as:
Total momentum after = (Your mass + Ball's mass) × (Your velocity after the collision)
Since the ball's velocity is opposite in direction, we will consider it negative:
Total momentum after = (71.5 kg + 0.405 kg) × (-v')
According to the principle of conservation of momentum:
Total momentum before = Total momentum after
So, we can set up the equation:
4.6575 kg·m/s = (71.5 kg + 0.405 kg) × (-v')
Now we can solve for v':
-v' = 4.6575 kg·m/s / (71.905 kg)
v' ≈ -0.0647 m/s
Therefore, after the collision, your speed (velocity) will be approximately 0.0647 m/s in the opposite direction of the ball.
Note: The negative sign in the velocity indicates the opposite direction.