A bullet of mass kg moving at 500 m⁄s embeds itself in a large, fixed piece of wood and travels 6 cm before coming to rest. Assuming that the deceleration of the bullet is constant, find the force exerted by the wood on the bullet.
To find the force exerted by the wood on the bullet, we can use Newton's second law of motion, which states that the force exerted on an object is equal to the mass multiplied by the acceleration.
Given:
Mass of the bullet (m) = kg
Initial velocity of the bullet (u) = 500 m/s
Final velocity of the bullet (v) = 0 m/s
Distance traveled by the bullet (s) = 6 cm = 0.06 m
First, we need to find the acceleration (a) of the bullet using the equation:
v^2 = u^2 + 2as
0 = (500^2) + 2a(0.06)
Simplifying the equation:
0 = 250000 + 0.12a
Rearranging:
0.12a = -250000
a = -250000 / 0.12
a ≈ -2,083,333.33 m/s^2 (since the acceleration is negative, indicating deceleration)
Now, we can calculate the force (F) exerted by the wood on the bullet using Newton's second law:
F = m * a
F = kg * -2,083,333.33 m/s^2
F ≈ -2,083,333.33 kg * m/s^2
Therefore, the force exerted by the wood on the bullet is approximately -2,083,333.33 kg * m/s^2. The negative sign indicates that the force is exerted in the opposite direction of the bullet's motion.
To find the force exerted by the wood on the bullet, we can use Newton's second law of motion, which states that force is equal to the rate of change of momentum.
First, let's find the initial momentum of the bullet. Momentum (p) is calculated by multiplying the mass (m) of the object by its velocity (v).
So, the initial momentum of the bullet is:
p_initial = m * v
= (mass of the bullet) * (velocity of the bullet)
= (mass) * 500
Next, let's find the final momentum of the bullet. Since the bullet comes to rest, its final momentum is zero.
p_final = 0
The change in momentum can be calculated by subtracting the final momentum from the initial momentum:
Δp = p_final - p_initial
= 0 - (mass * 500)
= - (mass * 500)
Now, we know that force (F) is equal to the rate of change of momentum (Δp) divided by the time (t) taken for the change to occur:
F = Δp / t
In this case, since the deceleration of the bullet is assumed to be constant, we can calculate the time taken (t) using the equation of motion:
s = ut + (1/2) * a * t^2,
where s is the displacement, u is the initial velocity, a is the acceleration, and t is the time.
The bullet travels a displacement of 6 cm before coming to rest, which is equal to 0.06 m. Since the bullet comes to rest, its final velocity is 0. Therefore, we can rewrite the equation as:
0.06 = 500t + (1/2) * (-a) * t^2
0.06 = 500t - 0.5at^2
Now, we know that the acceleration (a) is the rate of change of velocity (v) with respect to time (t). In this case, the final velocity (v) is 0, and the initial velocity (u) is 500 m/s. Therefore, we have:
a = (v - u) / t
= (0 - 500) / t
= -500 / t
Substituting this value of 'a' into the previous equation:
0.06 = 500t + 0.5 * (500 / t) * t^2
0.06 = 500t + 0.5 * 500 * t
0.06 = 500t + 250t
0.06 = 750t
t = 0.06 / 750
t ≈ 0.00008 s
Now we have the time (t), we can substitute this value back into the equation to calculate the force (F):
F = - (mass * 500) / t
= - (mass * 500) / 0.00008
So, to find the force exerted by the wood on the bullet, divide the product of the bullet mass and 500 by 0.00008.