A pitcher claims he can throw a 0.157 kg baseball with as much momentum as a 1.5 g bullet moving with a speed of 1.50 multiplied by 103 m/s.
(a) What must the baseball's speed be if the pitcher's claim is valid?
m/s
(b) Which has greater kinetic energy, the ball or the bullet?
To determine the speed of the baseball, we can use the principle of conservation of momentum. The momentum of an object can be calculated by multiplying its mass by its velocity.
(a) Let's assume the momentum of the bullet is equal to the momentum of the baseball.
The momentum of the bullet is given by:
momentum = mass * velocity
Given that the mass of the bullet is 1.5 g, which is equal to 1.5 * 10^-3 kg, and the velocity is 1.50 * 10^3 m/s, we can calculate the bullet's momentum as:
momentum of bullet = (1.5 * 10^-3 kg) * (1.50 * 10^3 m/s)
To achieve the same momentum, the baseball must have the same value. We can express this as:
(0.157 kg) * velocity of baseball = (1.5 * 10^-3 kg) * (1.50 * 10^3 m/s)
Simplifying the equation, we find the speed of the baseball:
velocity of baseball = ((1.5 * 10^-3 kg) * (1.50 * 10^3 m/s)) / (0.157 kg)
Now we can calculate the result:
velocity of baseball = 1.43 * 10^1 m/s
So, the speed of the baseball must be approximately 14.3 m/s.
(b) To determine which object has greater kinetic energy, we can use the kinetic energy formula:
kinetic energy = (1/2) * mass * velocity^2
Substituting the mass and velocity values for both the bullet and the baseball into the equation, we can compare their kinetic energies.
For the bullet:
kinetic energy of bullet = (1/2) * (1.5 * 10^-3 kg) * (1.50 * 10^3 m/s)^2
For the baseball:
kinetic energy of baseball = (1/2) * (0.157 kg) * (14.3 m/s)^2
Calculating the values, we find:
kinetic energy of bullet is approximately 1687.5 Joules
kinetic energy of baseball is approximately 157.3 Joules
Therefore, the bullet has a greater kinetic energy than the baseball.
(a) To determine the speed of the baseball, we can use the momentum formula:
Momentum = mass × velocity
The momentum of the bullet can be calculated as:
Momentum_bullet = mass_bullet × velocity_bullet
= 1.5 g × (1.50 × 10^3 m/s)
= 1.5 × 10^-3 kg × 1.50 × 10^3 m/s
According to the pitcher's claim, the momentum of the baseball is equal to that of the bullet:
Momentum_baseball = momentum_bullet
To find the speed of the baseball, we can rearrange the momentum formula:
Velocity_baseball = Momentum_baseball / mass_baseball
Since the mass of the baseball is given as 0.157 kg and the momentum is the same as the bullet, we substitute the values into the equation:
Velocity_baseball = (1.5 × 10^-3 kg × 1.50 × 10^3 m/s) / 0.157 kg
Simplifying the equation:
Velocity_baseball ≈ 14.35 m/s
Therefore, the speed of the baseball must be approximately 14.35 m/s for the pitcher's claim to be valid.
(b) The kinetic energy of an object can be calculated using the formula:
Kinetic energy = (1/2) × mass × velocity^2
To determine which has a greater kinetic energy, we compare the kinetic energies of the bullet and baseball.
Kinetic energy_bullet = (1/2) × mass_bullet × velocity_bullet^2
= (1/2) × 1.5 × 10^-3 kg × (1.50 × 10^3 m/s)^2
Kinetic energy_baseball = (1/2) × mass_baseball × velocity_baseball^2
= (1/2) × 0.157 kg × (14.35 m/s)^2
Evaluating these expressions:
Kinetic energy_bullet ≈ 1.69 J
Kinetic energy_baseball ≈ 14.52 J
From the calculations, it can be concluded that the baseball has a greater kinetic energy than the bullet.