A 69.5kg football player is gliding across very smooth ice at 2.15m/s . He throws a 0.470kg football straight forward.
A) What is the player's speed afterward if the ball is thrown at 16.0m/s relative to the ground?
B) What is the player's speed afterward if the ball is thrown at 16.0m/s relative to the player?
Solve using momentum
m1u1+m2u2=m1v1+m2v2
u=initial speed
v=final speed
m1,m2 masses
Using the ground as reference,
m1=69.5 kg
m2=0.470 kg
u1=u2=2.15 m/s
v2=16 m/s
m1u1+m2u2=m1v1+m2v2
Solve for v1 to get
v1=2.056 m/s approx.
B)
v2=16+2.15=18.15
Solve similarly to get 2.042 m/s approx.
A) Well, if the football player throws the ball at 16.0m/s relative to the ground, he better be prepared for some serious ice-skating! After the throw, his speed will be a combination of his initial speed and the speed of the throw. So, to find his speed afterward, we simply add the two speeds together.
Mathematically, Vafter = Vinitial + Vthrow
Given that the player's initial speed (Vinitial) is 2.15m/s, and the ball's speed relative to the ground (Vthrow) is 16.0m/s, we can substitute the values into the equation and calculate the answer:
Vafter = 2.15m/s + 16.0m/s
And voilà! The player's speed afterward will be the result of this sum.
B) If the ball is thrown at 16.0m/s relative to the player, we can still use the same equation as before to find his speed afterward. But this time, we need to subtract the speed of the throw from his initial speed. Again, let's plug in the values and calculate:
Vafter = Vinitial - Vthrow
Vafter = 2.15m/s - 16.0m/s
We'll need to take a moment here to appreciate the hilariously negative result. It seems like this throw has made our football player go backward in time! With a speed like that, he may need to enlist some physicists to understand the concept of negative velocity.
A) To solve this problem, we can apply the law of conservation of momentum. According to this law, the total momentum before the ball is thrown should be equal to the total momentum after the ball is thrown.
The initial momentum of the system is given by the equation:
Initial momentum = (mass of football player) x (initial speed of football player) + (mass of football) x (initial speed of football)
Plugging in the given values:
Initial momentum = (69.5 kg) x (2.15 m/s) + (0.470 kg) x (0 m/s) [Since the ball is initially at rest]
The final momentum of the system is given by the equation:
Final momentum = (mass of football player) x (final speed of football player) + (mass of football) x (final speed of football)
We are looking for the final speed of the football player, so we can rearrange the equation as:
Final speed of football player = (Final momentum - (mass of football) x (final speed of football)) / (mass of football player)
The final momentum of the system is:
Final momentum = (mass of football player) x (final speed of football player) + (mass of football) x (final speed of football)
Plugging in the given values:
(69.5 kg) x (2.15 m/s) + (0.470 kg) x (16.0 m/s) = (69.5 kg) x (final speed of football player) + (0.470 kg) x (16.0 m/s)
Now, we can rearrange the equation to solve for the final speed of the football player:
(69.5 kg) x (final speed of football player) = (69.5 kg) x (2.15 m/s) + (0.470 kg) x (16.0 m/s) - (0.470 kg) x (16.0 m/s)
Calculating the values, we get:
Final speed of football player = [ (69.5 kg) x (2.15 m/s) + (0.470 kg) x (16.0 m/s) - (0.470 kg) x (16.0 m/s) ] / (69.5 kg)
Simplifying the equation further:
Final speed of football player = [ (69.5 kg) x (2.15 m/s) ] / (69.5 kg)
After canceling out the similar units, we find:
Final speed of football player = 2.15 m/s
Therefore, the player's speed afterward, when the ball is thrown at 16.0 m/s relative to the ground, is 2.15 m/s.
B) When the ball is thrown at 16.0 m/s relative to the player, we can still apply the law of conservation of momentum. The initial momentum of the system remains the same.
The final momentum of the system is given by the equation:
Final momentum = (mass of football player) x (final speed of football player) + (mass of football) x (final speed of football)
Since the ball is thrown at 16.0 m/s relative to the player, we subtract this velocity from the initial speed of the football player when calculating the final speed of the football player.
Plugging in the given values:
Final momentum = (69.5 kg) x (final speed of football player) + (0.470 kg) x (0 m/s - 16.0 m/s)
Simplifying the equation:
Final momentum = (69.5 kg) x (final speed of football player) - (0.470 kg) x (16.0 m/s)
We can rearrange the equation to solve for the final speed of the football player:
(69.5 kg) x (final speed of football player) = (69.5 kg) x (2.15 m/s) - (0.470 kg) x (16.0 m/s)
Calculating the values, we find:
Final speed of football player = [ (69.5 kg) x (2.15 m/s) - (0.470 kg) x (16.0 m/s) ] / (69.5 kg)
Simplifying further:
Final speed of football player = [ (69.5 kg) x (2.15 m/s) ] / (69.5 kg) - [ (0.470 kg) x (16.0 m/s) ] / (69.5 kg)
After canceling out the similar units, we find:
Final speed of football player = 2.15 m/s - 0.470 m/s
Therefore, the player's speed afterward, when the ball is thrown at 16.0 m/s relative to the player, is approximately 1.68 m/s.
To solve this problem, we can apply the law of conservation of momentum. According to this law, the total momentum before an event is equal to the total momentum afterward. In this case, the total momentum before the throw is the sum of the player's momentum and the ball's momentum.
The formula for momentum is given by:
momentum = mass × velocity
Let's first calculate the player's momentum before the throw. The player has a mass of 69.5 kg and a speed of 2.15 m/s, so his momentum is:
player's momentum before = 69.5 kg × 2.15 m/s
Now let's calculate the ball's momentum before the throw. The ball has a mass of 0.470 kg and is initially at rest relative to the player, so its momentum before the throw is:
ball's momentum before = 0.470 kg × 0 m/s
Since the player and the ball are initially at rest relative to each other, the total momentum before the throw is:
total momentum before = player's momentum before + ball's momentum before
Now let's consider the two scenarios.
A) What is the player's speed afterward if the ball is thrown at 16.0 m/s relative to the ground?
In this scenario, we need to consider the ball's momentum relative to the ground. The ball's momentum after the throw can be calculated as:
ball's momentum after = 0.470 kg × 16.0 m/s
Since the total momentum before the throw is equal to the total momentum after the throw, we can equate the two expressions:
total momentum before = total momentum after
(player's momentum before + ball's momentum before) = (player's momentum after + ball's momentum after)
Since the ball's momentum after the throw is non-zero, the player's momentum after the throw must also be non-zero. Therefore, the player's speed afterward will be greater than 0 m/s.
To find the player's speed afterward, we need to isolate the player's momentum after and divide it by the player's mass:
player's momentum after = (total momentum before - ball's momentum after)
player's speed afterward = player's momentum after / player's mass
Substitute the values we calculated earlier to find the answer.
B) What is the player's speed afterward if the ball is thrown at 16.0 m/s relative to the player?
In this case, the ball's momentum after the throw is relative to the player. Since the ball is thrown in the same direction as the player's initial motion, the ball's momentum after the throw will add to the player's momentum. This will result in an increase in his speed relative to the ground.
Using the same approach as in scenario A, calculate the player's momentum after and divide it by his mass to find the player's speed afterward.
Remember to substitute the appropriate values and units to get the final answer.