8. What average force is needed to accelerate a 7.00-gram pellet from rest to 125 m/s over a distance of 0.800 m along the barrel of a rifle?
vf^2= 2ad = 2 (force/mass) d
force= vf^2*mass/2d
To find the average force needed to accelerate the pellet, we can use Newton's second law of motion, which states that force is equal to mass multiplied by acceleration (F = m * a). In this case, we are given the mass of the pellet (7.00 grams) and the final velocity (125 m/s), but we need to find the acceleration.
To find the acceleration, we can use the equations of motion. The first equation is v = u + at, where v is the final velocity, u is the initial velocity (which is 0 in this case since the pellet is at rest), a is the acceleration, and t is the time taken. From this equation, we can rearrange it to find t:
t = (v - u) / a
Since the pellet is at rest initially, u = 0, so the equation simplifies to t = v / a.
The second equation of motion is s = ut + (1/2)at^2, where s is the displacement (0.800 m in this case), u is the initial velocity (0), a is the acceleration, and t is the time. Rearranging this equation to solve for a:
a = 2(s - ut) / t^2
Substituting the values we have:
a = 2(0.800 - 0 * t) / t^2
a = 1.6 / t^2
Now we can substitute the value of a into Newton's second law equation to find the average force:
F = m * a
F = (7.00 g) * (1.6 / t^2)
To calculate the average force, we need to find the value of t. We already have the final velocity, 125 m/s, and we know the distance traveled along the barrel, 0.800 m. We can use the equation v = s / t to find the time:
t = s / v
t = 0.800 m / 125 m/s
Now we can substitute this value of t into the equation for average force and solve for F:
F = (7.00 g) * (1.6 / (0.800 m / 125 m/s)^2)
First, we need to convert the mass from grams to kilograms, since the SI unit for measuring force is Newtons (kg * m/s^2):
F = (0.007 kg) * (1.6 / (0.800 m / 125 m/s)^2)
Calculating this expression will give us the average force needed to accelerate the pellet.