A proton (mass m = 1.67 10-27 kg) is being accelerated along a straight line at 2.1 1015 m/s2 in a machine. The proton has an initial speed of 2.4 107 m/s and travels 4.2 cm.
(a) What is its speed?
1 . m/s
(b) What is the increase in its kinetic energy?
2 . J
20 m/s
2 J
To find the speed of the proton, you need to use the formula for acceleration:
acceleration = change in velocity / time
In this case, the acceleration is given as 2.1 * 10^15 m/s^2. Let's assume the time taken by the proton to travel the given distance is 't'.
Now, the change in velocity can be calculated using the equation of motion:
change in velocity = initial velocity + acceleration * time
The initial velocity is given as 2.4 * 10^7 m/s.
Setting up the equation:
2.1 * 10^15 m/s^2 = (change in velocity) / t
Solving for change in velocity:
change in velocity = 2.1 * 10^15 m/s^2 * t
Substituting the given distance and solving for time:
4.2 cm = (2.4 * 10^7 m/s * t) + (0.5 * 2.1 * 10^15 m/s^2 * t^2)
Converting the distance to meters:
4.2 cm = 0.042 m
0.042 m = (2.4 * 10^7 m/s * t) + (0.5 * 2.1 * 10^15 m/s^2 * t^2)
Now, you can solve this quadratic equation for 't'. Plugging the values into the quadratic equation:
0.5 * 2.1 * 10^15 t^2 + 2.4 * 10^7 t - 0.042 = 0
Using the quadratic formula, you can solve for 't':
t = (-b ± sqrt(b^2 - 4ac)) / (2a)
In this case, a = 0.5 * 2.1 * 10^15, b = 2.4 * 10^7, and c = -0.042.
After solving for 't', you can use it to find the change in velocity and then calculate the final speed using the equation:
final speed = initial speed + change in velocity
To find the increase in kinetic energy, you need to use the formula:
change in kinetic energy = 0.5 * mass * (final speed^2 - initial speed^2)
In this case, the mass of the proton is given as 1.67 * 10^-27 kg. Plugging in the values, you can calculate the change in kinetic energy.