A 0.450 kg hammer is moving horizontally at 8.00 m/s when it strikes a nail and comes to rest after driving it 1.00 cm into a board.(a) Calculate the duration of the impact.(b) What was the average force exerted on the nail?
To solve this problem, we can use the principles of conservation of momentum and work-energy theorem.
(a) To calculate the duration of the impact, we need to determine the time it takes for the hammer to come to rest when it strikes the nail. This can be done using the equation of motion:
V^2 = U^2 + 2as
Where:
V is the final velocity (0 m/s, since the hammer comes to rest)
U is the initial velocity (8.00 m/s)
a is the acceleration
s is the displacement (1.00 cm = 0.01 m)
Rearranging the equation, we get:
a = (V^2 - U^2) / (2s)
Substituting the given values, we have:
a = (0^2 - (8.00 m/s)^2) / (2 * 0.01 m)
Simplifying:
a = -64.00 m^2/s^2 / 0.02 m
a = -3200 m/s^2
Now, we can use another equation of motion:
V = U + at
Substituting the known values:
0 = 8.00 m/s + (-3200 m/s^2)t
Solving for t:
-8.00 m/s = -3200 m/s^2 * t
t = -8.00 m/s / -3200 m/s^2
t = 0.0025 s
Therefore, the duration of the impact is 0.0025 seconds.
(b) To calculate the average force exerted on the nail, we can use the work-energy theorem. The work done by the hammer on the nail is equal to the change in kinetic energy of the hammer. The work done can be calculated using the equation:
Work = Force * Distance * cos(θ)
In this case, the hammer was moving horizontally, so the angle θ between the force and displacement vectors is 0 degrees (cos(0) = 1).
The work done is given by:
Work = Change in kinetic energy
The initial kinetic energy of the hammer is given by:
KE_initial = (1/2) * m * U^2
Substituting the known values:
KE_initial = (1/2) * 0.450 kg * (8.00 m/s)^2
KE_initial = 14.40 J
The final kinetic energy of the hammer is zero since it comes to rest.
Therefore, the change in kinetic energy is:
ΔKE = KE_final - KE_initial
= 0 J - 14.40 J
= -14.40 J
Since work equals the change in kinetic energy:
Work = -14.40 J
Substituting this into the work equation:
-14.40 J = Force * 0.01 m * 1
Simplifying:
Force = -14.40 J / (0.01 m)
Force = -1440 N
The negative sign indicates that the force is in the opposite direction of the displacement. However, since force is a scalar quantity, we take the magnitude of the force:
Average Force = 1440 N
Therefore, the average force exerted on the nail is 1440 Newtons.