Suppose that you have a masa os 45.7 kg and are standing on frictionless roller skates. Someone then throws you a 2.50 kg mass with a velocity of 14.5 m/s and you catch it. What will be your resultant velocity?
Conservation of Momentum:
m1u1+m2u2=m1v1+m2v2
m=mass
u=initial velocity
v=velocity after impact
1,2 objects, people.
Since the person catches the mass, the common velocity after "impact" is
v1=v2=v.
Solve for v, since all other variables are known.
To find the resultant velocity after catching the mass, we can use the principle of conservation of momentum. According to this principle, the total momentum before the catch should be equal to the total momentum after the catch.
The momentum of an object can be calculated by multiplying its mass and velocity. So, before the catch, the total momentum is the sum of your momentum (m1) and the momentum of the thrown mass (m2), while after the catch, it is the sum of your new momentum (m1'), with the added mass, and the momentum of the thrown mass after being caught (m2').
Mathematically, we can represent this as:
m1 * v1 + m2 * v2 = m1' * v1' + m2' * v2'
where:
m1 = your mass = 45.7 kg
v1 = your initial velocity = 0 m/s (since you were initially at rest)
m2 = mass thrown = 2.50 kg
v2 = velocity of the thrown mass = 14.5 m/s
v1' = your resultant velocity (after the catch)
m2' = mass of the caught mass = 2.50 kg
Using this equation, we can solve for v1':
45.7 kg * 0 m/s + 2.50 kg * 14.5 m/s = 45.7 kg * v1' + 2.50 kg * v2'
0 kg·m/s + 36.25 kg·m/s = 45.7 kg·v1' + 2.50 kg·14.5 m/s
36.25 kg·m/s = 45.7 kg·v1' + 36.25 kg·m/s
Subtracting 36.25 kg·m/s from both sides:
0 = 45.7 kg·v1'
Dividing both sides by 45.7 kg:
v1' = 0 m/s
Therefore, your resultant velocity after catching the mass will be 0 m/s.
To find the resultant velocity, we can use the principle of conservation of momentum. According to this principle, the total momentum before the catch is equal to the total momentum after the catch.
The equation for momentum is:
momentum = mass × velocity
Let's calculate the momentum before and after the catch:
Before the catch:
Momentum of you = mass of you × velocity of you
= 45.7 kg × 0 m/s (since you are at rest)
= 0 kg·m/s
Momentum of the thrown mass = mass of the thrown mass × velocity of the thrown mass
= 2.50 kg × 14.5 m/s
= 36.25 kg·m/s
Total momentum before the catch = Momentum of you + Momentum of the thrown mass
= 0 kg·m/s + 36.25 kg·m/s
= 36.25 kg·m/s
After the catch:
Since both you and the thrown mass are now moving together with a common velocity, let's call it v (resultant velocity). Therefore, the resulting momentum can be calculated as follows:
Momentum of you and the thrown mass = (mass of you + mass of the thrown mass) × resultant velocity
= (45.7 kg + 2.50 kg) × v
= 48.2 kg × v
Since the total momentum before and after the catch is the same as per the principle of conservation of momentum, we have:
Total momentum before the catch = Total momentum after the catch
Therefore, we can write:
36.25 kg·m/s = 48.2 kg × v
Now, let's solve for v by rearranging the equation:
v = (36.25 kg·m/s) / 48.2 kg
= 0.752 m/s (rounded to three decimal places)
Therefore, your resultant velocity after catching the thrown mass will be approximately 0.752 m/s.