An 8.4 g bullet is accelerated in a rifle barrel 45 cm long to a speed of 1020 m/s.
Use the work-energy theorem to find the
average force exerted on the bullet while it is being accelerated.
To find the average force exerted on the bullet while it is being accelerated, we can use the work-energy theorem. According to the theorem, the work done on an object is equal to the change in its kinetic energy.
First, let's find the initial kinetic energy of the bullet. The formula for kinetic energy is:
KE = (1/2)mv^2
Where:
KE = kinetic energy
m = mass of the bullet
v = velocity of the bullet
Given:
m = 8.4 g = 0.0084 kg
v = 1020 m/s
Plugging in these values into the formula, we get:
KE = (1/2)(0.0084 kg)(1020 m/s)^2 = 431.2728 J (Joules)
The work done on the bullet is equal to its change in kinetic energy. Since the bullet starts from rest and accelerates to a speed of 1020 m/s, the change in kinetic energy is:
ΔKE = KE_final - KE_initial = (1/2)mv_final^2 - (1/2)mv_initial^2
Now, for the final kinetic energy, we can use the same formula since the velocity is already given:
KE_final = (1/2)mv_final^2 = (1/2)(0.0084 kg)(1020 m/s)^2 = 431.2728 J (Joules)
The initial kinetic energy is zero because the bullet starts from rest.
ΔKE = 431.2728 J - 0 J = 431.2728 J
According to the work-energy theorem, this change in kinetic energy is equal to the work done on the bullet. Therefore, the work done on the bullet is 431.2728 J.
Now, to find the average force exerted on the bullet, we can use the formula:
Work = Force * Distance
Since the work done on the bullet is 431.2728 J and the distance traveled by the bullet in the barrel is given as 45 cm (or 0.45 m), we can substitute these values into the formula:
431.2728 J = Force * 0.45 m
Solving for force, we get:
Force = 431.2728 J / 0.45 m = 957.272 J/m ≈ 957.3 N/m
Therefore, the average force exerted on the bullet while it is being accelerated is approximately 957.3 Newtons.
Work = Force * (barrel length) = final kinetic energy
Solve for the Force