A 10-kg mass is moving at a speed of 4.0 m/s. How much work is required to stop the mass?
A. 20 J
B. 40 J
C. 60 J
D. 80 J
E. 100 J
80j
To calculate the amount of work required to stop the mass, we need to use the formula for work, which is given by W = F * d * cos(theta), where W is work, F is the force applied, d is the displacement, and theta is the angle between the force and displacement vectors.
In this case, the force applied to the mass would be the force required to bring the mass to rest from its initial velocity of 4.0 m/s. This force is equal to the rate of change of momentum, which can be calculated using the formula F = m * a, where m is the mass and a is the acceleration.
To stop the mass, the acceleration would be equal to the negative of its initial velocity divided by the time taken to stop. So, a = (-4.0 m/s) / t, where t is the time taken to stop.
Since we want to calculate the work required to bring the mass to rest from a given velocity, we need to find the displacement, which is the distance traveled during the time taken to stop. The distance traveled can be calculated using the formula d = (1/2) * a * t^2.
Plugging in the values, we have F = (10 kg) * [(-4.0 m/s) / t], and d = (1/2) * [(-4.0 m/s) / t] * t^2.
Now, we can substitute these values into the work formula:
W = F * d * cos(theta)
W = (10 kg) * [(-4.0 m/s) / t] * (1/2) * [(-4.0 m/s) / t] * t^2 * cos(theta)
Since the mass is brought to rest, the angle between the force and displacement vectors would be 0 degrees, which means cos(theta) = 1. Therefore, we ignore the cos(theta) term.
W = (10 kg) * [(-4.0 m/s) / t] * (1/2) * [(-4.0 m/s) / t] * t^2
W = (10 kg) * (1/2) * [(-4.0 m/s) / t] * [(-4.0 m/s) / t] * t^2
Simplifying further, we have:
W = (10 kg) * (1/2) * (16.0 m^2/s^2) * t
W = 80.0 kg * m^2/s^2 * t
Now, we need to find the value of time, t. We know that the mass is brought to rest, so the final velocity is 0 m/s. We can use the formula for velocity, v = u + a * t, where u is the initial velocity, v is the final velocity, a is the acceleration, and t is the time taken to reach the final velocity.
Plugging in the values, we have:
0 m/s = 4.0 m/s + (-4.0 m/s^2) * t
Solving for t, we find:
-4.0 m/s^2 * t = -4.0 m/s
t = 1 s
Now, we can substitute the value of t into the work equation:
W = 80.0 kg * m^2/s^2 * 1 s
W = 80.0 kg * m^2/s^2
Therefore, the amount of work required to stop the mass is 80 J, which corresponds to option D.