A hammer weighing of 1 kg moving with the speed of 10 m/s strikes the head of the nail driving it 10 cm into wall. Neglecting the mass of the nail, calculate:
1) The acceleration during the impact
2) The time interval during the impact
3) The impulse (force X time)
To answer your questions, let's break down the problem step by step:
1) The acceleration during the impact:
Acceleration (a) can be calculated using the equation:
a = (vf - vi) / t
where vf is the final velocity, vi is the initial velocity, and t is the time interval.
The initial velocity of the hammer is 10 m/s, and it comes to rest after striking the nail, so the final velocity is 0 m/s. Let's calculate the acceleration:
a = (0 - 10 m/s) / t
Since the velocity changed from positive to zero, we can say that the direction of acceleration is opposite to the initial velocity. Therefore, the acceleration is negative.
2) The time interval during the impact:
The time interval (t) can be calculated using the equation:
d = vit + (1/2)at^2
where d is the displacement.
The displacement (d) is given as 10 cm, which is equal to 0.1 m. The initial velocity (vi) is 10 m/s, and the acceleration (a) is calculated as -10 m/s^2 (using the equation from step 1).
0.1 m = (10 m/s) * t + (1/2) * (-10 m/s^2) * t^2
Simplifying the equation, we get:
0 = 10t - 5t^2
Rearranging the equation, we get:
5t^2 - 10t = 0
t(5t - 10) = 0
t = 0 (rejecting this solution since it doesn't make sense in this context) or t = 2 seconds
Therefore, the time interval during the impact is 2 seconds.
3) The impulse (force X time):
Impulse can be calculated using the equation:
Impulse = Force * Time
In this case, since the mass of the nail is neglected, and impulse is equal to the change in momentum.
Momentum (p) can be calculated using the equation:
p = m * v
where m is the mass and v is the velocity.
The mass (m) of the hammer is given as 1 kg, and the initial velocity (vi) is 10 m/s. So the initial momentum (pi) is:
pi = (1 kg) * (10 m/s) = 10 kg·m/s
After striking the nail, the hammer comes to rest, so the final momentum (pf) is:
pf = 0 kg·m/s
Therefore, the impulse can be calculated as:
Impulse = pf - pi = 0 kg·m/s - 10 kg·m/s = -10 kg·m/s
So, the impulse during the impact is -10 kg·m/s.
To answer these questions, we will use the principles of linear motion and conservation of momentum. Here's how you can calculate each value:
1) The acceleration during the impact:
Acceleration can be calculated using the equation:
Acceleration = change in velocity / time interval
In this case, the change in velocity is given by the initial velocity (10 m/s) minus the final velocity (0 m/s, as the hammer stops moving after impact). We need to convert 10 cm to meters by dividing by 100. The time interval is what we need to find.
So, we have:
Initial velocity = 10 m/s
Final velocity = 0 m/s
Change in velocity = Initial velocity - Final velocity
Change in velocity = 10 m/s - 0 m/s = 10 m/s
Change in velocity = Acceleration * time interval
10 m/s = Acceleration * time interval
Solving for acceleration:
Acceleration = 10 m/s / time interval
2) The time interval during the impact:
We can use the equation derived above to find the time interval:
10 m/s = Acceleration * time interval
From the previous equation, we know that acceleration = 10 m/s divided by time interval. Substitute this value in:
10 m/s = (10 m/s) / time interval
Multiply both sides by time interval to solve for it:
10 m/s * time interval = 10 m/s
Divide both sides by 10 m/s:
time interval = 1 second
Therefore, the time interval during the impact is 1 second.
3) The impulse (force x time):
Impulse is the change in momentum of an object and can be calculated using the equation:
Impulse = force x time
The force can be calculated using Newton's second law of motion:
Force = mass x acceleration
In this case, the mass of the hammer is given as 1 kg, and the acceleration can be calculated using the equation derived in question 1.
Acceleration = 10 m/s / time interval
Acceleration = 10 m/s / 1 s
Acceleration = 10 m/s^2
Substitute the values in Newton's second law equation:
Force = 1 kg x 10 m/s^2
Force = 10 N
Now we can calculate the impulse:
Impulse = force x time
Impulse = 10 N x 1 s
Impulse = 10 Ns
Therefore, the impulse during the impact is 10 Ns.
well, 3 is the easiest of all. Thw impulse is the change of momentum.MV = 1 *10 = 10 kg m/s
average speed during stop = 10/2 = 5 m/s
so
t = time to stop = 0.10 meter/ 5 meters/s = 0.02 second. That is number 2
a = change in v/ time = -10/.02 = -500 m/s^2
that is number 1.
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now you could have one it all this way:
d = Vi t + .5 a t^2
v = Vi + a t
so
0 = 10 + a t
a t = -10
0.10 = 10 t + .5 (-10)t
.1 = 5 t
t = .02 seconds which we knew
then
a t = a(.02)= -10
so a = -500 m/s^2