If an 75-kg baseball pitcher wearing frictionless roller skates picks up a 0.145-kg baseball and pitches it toward the south at 40 m/s, how fast will he begin moving toward the north?
Well, I'm no expert in physics, but I'm guessing that if the pitcher is wearing frictionless roller skates, their movement will result from the law of conservation of momentum. So, if the baseball is pitched towards the south, then (using my excellent math skills) the pitcher will begin moving towards the north at the same speed as the baseball but with opposite direction. So, the pitcher will start moving towards the north at a speedy 40 m/s! Just don't forget to warn him about any incoming walls or obstacles!
To determine the speed at which the baseball pitcher will begin moving toward the north, we can use the principle of conservation of momentum. According to this principle, the total momentum before the pitch must be equal to the total momentum after the pitch.
The initial momentum is the sum of the momenta of the pitcher and the baseball.
The momentum (p) of an object is given by the product of its mass (m) and velocity (v): p = mv.
Given:
Mass of the pitcher (m1) = 75 kg
Velocity of the pitcher (v1) = 0 m/s (since the pitcher is initially at rest)
Mass of the baseball (m2) = 0.145 kg
Velocity of the baseball (v2) = -40 m/s (toward the south)
Using the conservation of momentum, we can write the equation:
(m1 * v1) + (m2 * v2) = (m1 * v1') + (m2 * v2')
where v1' is the velocity of the pitcher after the pitch and v2' is the velocity of the baseball after the pitch.
Substituting the values into the equation:
(75 kg * 0 m/s) + (0.145 kg * -40 m/s) = (75 kg * v1') + (0.145 kg * v2')
Simplifying the equation:
-5.8 kg⋅m/s = 75 kg⋅v1' + 0.145 kg⋅v2'
Since the pitcher is initially at rest (v1 = 0 m/s), the equation becomes:
-5.8 kg⋅m/s = 75 kg⋅v1' + 0.145 kg⋅v2'
Solving for v1':
75 kg⋅v1' = -5.8 kg⋅m/s - (0.145 kg⋅v2')
v1' = (-5.8 kg⋅m/s - (0.145 kg⋅v2')) / 75 kg
Substituting the value of v2' = -40 m/s:
v1' = (-5.8 kg⋅m/s - (0.145 kg⋅(-40 m/s))) / 75 kg
Calculating the value:
v1' = (-5.8 kg⋅m/s + 5.8 kg⋅m/s) / 75 kg
v1' = 0 m/s
Hence, the baseball pitcher will not begin moving at all toward the north after pitching the baseball.
To find the speed at which the baseball pitcher will begin moving toward the north, we can make use of the law of conservation of momentum. According to this law, the total momentum of an isolated system remains constant before and after an event.
The initial momentum of the system is given by the momentum of the baseball pitcher, which can be calculated as the product of his mass and velocity. The final momentum of the system is the sum of the momentum of the baseball pitcher and the momentum of the baseball after it is pitched.
Let's break down the solution step by step:
Step 1: Calculate the initial momentum of the system.
Initial momentum of the pitcher = (mass of pitcher) * (velocity of pitcher)
= 75 kg * 0 m/s (since the pitcher is initially at rest)
= 0 kg·m/s
Step 2: Calculate the momentum of the baseball after it is pitched.
Momentum of the baseball = (mass of baseball) * (velocity of baseball)
= 0.145 kg * (-40 m/s) (since the baseball is pitched towards the south)
= -5.8 kg·m/s
(Note: The negative sign indicates the direction of momentum, with a negative value indicating motion in the opposite direction)
Step 3: Calculate the final momentum of the system.
Final momentum of the system = Initial momentum of the system + Momentum of the baseball
= 0 kg·m/s + (-5.8 kg·m/s)
= -5.8 kg·m/s
Step 4: Determine the final velocity of the pitcher.
Final velocity of the pitcher = (Final momentum of the system) / (mass of the pitcher)
= (-5.8 kg·m/s) / (75 kg)
≈ -0.0773 m/s
Therefore, the baseball pitcher will begin moving toward the north at approximately 0.0773 m/s.