Physics Problem – Force on a Bullet in a Rifle
Question:
A 5.00 g bullet is fired from a rifle. As it exits the 50.0 cm long barrel, it has a velocity of 200 m/s.
(a) Use Newton’s Laws and the kinematic equations to determine the force on the bullet from the expanding gases as it travels down the barrel.
(b) Use the Work-Energy Theorem to determine the force on the bullet from the expanding gases as it travels down the barrel.
Answer with Full Detailed Explanation:
Part (a) Using Newton’s Laws and Kinematic Equations
Given:
Mass of bullet: m = 5.00 g = 5.00 × 10⁻³ kg
Initial velocity: u = 0 m/s (starts from rest)
Final velocity: v = 200 m/s
Distance (length of barrel): s = 50.0 cm = 0.5 m
Mass of bullet: m = 5.00 g = 5.00 × 10⁻³ kg
Initial velocity: u = 0 m/s (starts from rest)
Final velocity: v = 200 m/s
Distance (length of barrel): s = 50.0 cm = 0.5 m
Step 1: Find acceleration using kinematic equation
v² = u² + 2as
(200)² = 0 + 2 × a × 0.5
40000 = a
a = 4.0 × 10⁴ m/s²
Step 2: Use Newton’s second law
F = ma
F = (5.0 × 10⁻³ kg) × (4.0 × 10⁴ m/s²)
F = 200 N
(a) The force on the bullet using Newton’s laws is 200 N.
Part (b) Using the Work-Energy Theorem
The Work-Energy Theorem states:
W = ΔK = Kf – Ki
Since the bullet starts from rest, Ki = 0
Step 1: Final Kinetic Energy
Kf = (1/2)mv² = (1/2)(5.0 × 10⁻³)(200)²
Kf = 0.5 × 5.0 × 10⁻³ × 40000 = 100 J
Step 2: Work Done = Force × Distance
W = F × s
100 = F × 0.5
F = 200 N
(b) The force on the bullet using the Work-Energy Theorem is also 200 N.