A proton is moving at 425 m/s. (a) How much work must be done on it to stop it?(answer in joules) (A proton has a mass of 1.67×10−27 kg.) (b) Assume the net braking force acting on it has magnitude 8.01×10−16 N and is directed opposite to its initial velocity. Over what distance must the force be applied(answer in meters)? Watch your negative signs in this problem.
To solve these problems, we need to use the concepts of work and force. Let's break down each part of the question:
(a) How much work must be done on the proton to stop it?
Work (W) is defined as the product of force (F) applied on an object and the distance (d) over which the force is applied. In this case, the force applied is the net braking force acting on the proton.
To find the work done:
W = F * d
The given mass of the proton is 1.67×10^−27 kg. The proton is moving at a velocity of 425 m/s. To stop it, a force needs to be applied in the opposite direction of its velocity until its velocity becomes zero.
The force required to stop the proton is equal to the rate of change of momentum. Using the equation: F = ma, where 'm' is the mass of the proton and 'a' is the acceleration.
The acceleration (a) can be calculated using the equation: a = (v_final - v_initial) / t, where v_final is the final velocity (which is 0 in this case), v_initial is the initial velocity, and t is the time taken to stop the proton.
Given that v_initial = 425 m/s, v_final = 0 m/s, and we need to find the time taken to stop the proton (t).
Since v_final = v_initial + at, we can rearrange the equation to solve for t:
t = (v_final - v_initial) / a
Now, we substitute the known values into the equation:
t = (0 - 425) / a
The next step is to calculate the acceleration (a) using Newton's second law:
F = ma
a = F / m
Given that F = 8.01×10^−16 N and m = 1.67×10^−27 kg, we can calculate the acceleration.
a = (8.01×10^−16 N) / (1.67×10^−27 kg)
Now that we have the acceleration, we can substitute it back into the equation to find the time taken to stop the proton (t).
t = (0 - 425) / a
Once we have the time taken (t), we can now find the work done on the proton using the equation W = F * d.
(b) Over what distance must the force be applied to stop the proton?
In this case, we need to find the distance (d) over which the force is applied using the equation:
d = (v_initial * t) + (0.5 * a * t^2)
Substituting the known values for v_initial, t, and a, we can find the distance required.
By following these steps, you should be able to calculate the work done and the distance over which the force must be applied.