A 2.0 g bullet leaves the barrel of a gun at a speed of 260 m/s.
(a) Find its kinetic energy.
J
(b) Find the average force exerted by the expanding gases on the bullet as the bullet moves the length of the 43 cm long barrel.
N
(a) Well, a bullet traveling at high speed certainly has some kinetic energy. To find it, we can use the formula for kinetic energy: K.E. = (1/2) * mass * velocity^2. Plugging in the values: mass = 2.0 g = 0.002 kg (since 1 g = 0.001 kg), and velocity = 260 m/s, we get:
K.E. = (1/2) * 0.002 kg * (260 m/s)^2
= 0.002 kg * 33800 m^2/s^2
= 67.6 J
So, the kinetic energy of the bullet is 67.6 Joules.
(b) Now, let's calculate the average force exerted by the expanding gases on the bullet. To do that, we can use the equation F = Δp / Δt, where F is the force, Δp is the change in momentum, and Δt is the change in time.
First, let's find the change in momentum. The initial momentum of the bullet is given by p = m * v, where m is the mass of the bullet and v is its velocity. Plugging in the values: mass = 2.0 g = 0.002 kg, and velocity = 260 m/s, we get:
Initial momentum = 0.002 kg * 260 m/s
= 0.52 kg*m/s
Next, we need to find the final momentum of the bullet. Since the bullet leaves the barrel and there are no external forces acting on it, the final momentum is simply the initial momentum. Therefore, the change in momentum is:
Change in momentum = Final momentum - Initial momentum
= 0.52 kg*m/s - 0.52 kg*m/s
= 0 kg*m/s
Now, let's calculate the change in time. We know that the bullet moves the length of the 43 cm long barrel. Converting that to meters: 43 cm = 0.43 m.
Since we're looking for an average force, we need to find the average time it takes for the bullet to move the length of the barrel. Let's assume it takes t seconds. Therefore, the change in time is:
Change in time = t - 0
= t seconds
Finally, we can calculate the average force using the formula F = Δp / Δt:
Average force = 0 kg*m/s / t seconds
= 0 N
It seems like the average force exerted by the expanding gases on the bullet over the length of the barrel is zero. Maybe the bullet had some help from a little magic?
To answer these questions, we'll need to use the formulas for kinetic energy and average force.
(a) The formula for kinetic energy is given by the equation:
Kinetic energy (KE) = (1/2) * mass * velocity^2
Given that the mass (m) of the bullet is 2.0 g (or 0.002 kg) and the velocity (v) is 260 m/s, we can calculate the kinetic energy as follows:
KE = (1/2) * 0.002 kg * (260 m/s)^2
= 0.002 kg * 33800 m^2/s^2
= 67.6 J
Therefore, the kinetic energy of the bullet is 67.6 J.
(b) The formula for average force (F) is given by the equation:
Average force (F) = change in momentum / time
The change in momentum is equal to the final momentum (mv) minus the initial momentum (0, since the bullet starts from rest). The final momentum is given by the equation:
Final momentum = mass * velocity
Therefore, the change in momentum is given by:
Change in momentum = mass * velocity - 0
= mass * velocity
= 0.002 kg * 260 m/s
= 0.52 kg·m/s
Since the time (t) and length (d) are given by the problem, we can calculate the average force as follows:
Average force = (change in momentum) / (time)
= (0.52 kg·m/s) / (43 cm * 0.01 m/cm)
= 12.09 N
Therefore, the average force exerted by the expanding gases on the bullet is 12.09 N.
To find the kinetic energy of the bullet, we can use the formula:
Kinetic Energy = (1/2) * mass * velocity^2
Given that the mass of the bullet is 2.0 g, which is equivalent to 0.002 kg, and the velocity is 260 m/s, we can substitute these values into the formula to find the answer.
Kinetic Energy = (1/2) * 0.002 kg * (260 m/s)^2
Calculating this, we get:
Kinetic Energy = (1/2) * 0.002 kg * (67600 m^2/s^2) = 68.0 Joules (J)
So, the kinetic energy of the bullet is 68.0 J.
Now, let's move on to finding the average force exerted by the expanding gases on the bullet as it moves through the barrel.
To calculate the average force, we can use Newton's second law of motion, which states:
Force = mass * acceleration
Given that mass is 0.002 kg and the length of the barrel is 43 cm, which is equivalent to 0.43 m, we need to find the acceleration of the bullet.
Using the equation of motion, v^2 = u^2 + 2as, where v is the final velocity, u is the initial velocity, a is the acceleration, and s is the distance traveled, we can rearrange it to solve for acceleration:
a = (v^2 - u^2) / (2s)
Substituting the given values, we get:
a = [(260 m/s)^2 - 0^2] / (2 * 0.43 m) = (67600 m^2/s^2) / 0.86 m = 78604.65 m^2/s^2
Next, we can calculate the force:
Force = mass * acceleration = 0.002 kg * 78604.65 m^2/s^2
Calculating this, we get:
Force = 157.21 Newtons (N)
So, the average force exerted by the expanding gases on the bullet as it moves through the 43 cm long barrel is 157.21 N.
(a) You must have seen the definition of kinetic energy, which is
(1/2) M V^2
You have been told the mass M and the velocity V. Just do the calculation. The mass will have to be in kg if you want the answer in joules.
(b) Final kinetic energy = (average force)*(distance over which the force is applied)
(1/2)(0.002 kg)*(260 m/s)^2 = Fav*(0.43 m)
In your case, the distance is the gun barrel length. Distance must be in meters to get the force in Joules.
Solve the above equation for Fav, the average force.