A proton (mass ) is being accelerated along a straight line at in a machine. If the proton has an initial speed of and travels 3.5 cm, what then is (a) its speed and (b) the increase in its kinetic energy?
You need to specify the acceleration and also provide the missing initial speed following the word "of".
You will also need the mass of the proton, which you can always look up.
To find the speed and the increase in kinetic energy of the proton, we'll use the principles of uniform acceleration.
(a) To find the speed of the proton after traveling 3.5 cm, we'll start by using the formula of motion:
s = ut + 0.5 at^2,
where:
- s is the displacement (3.5 cm),
- u is the initial velocity (0.75 × 10^7 m/s),
- a is the acceleration (4.0 × 10^14 m/s²),
- t is the time (which we need to find).
First, we need to convert the displacement from centimeters to meters, since the other quantities are in SI units:
s = 3.5 cm = 0.035 m.
Now, we rearrange the formula to solve for time:
t = (-u + sqrt(u^2 + 2as))/a.
Plugging in the values, we get:
t = (-0.75 × 10^7 + sqrt((0.75 × 10^7)^2 + 2 × 4.0 × 10^14 × 0.035)) / 4.0 × 10^14.
Evaluating this expression will give us the time taken by the proton.
Once we have the time, we can calculate the speed by using the formula:
v = u + at.
Simply plug in the values of u (initial velocity) and t (time) to get the final velocity.
(b) To find the increase in kinetic energy, we use the formula:
ΔKE = KE_final - KE_initial.
The initial kinetic energy, KE_initial, can be calculated using the formula:
KE_initial = 0.5 × m × u^2,
where m is the mass of the proton and u is the initial velocity.
The final kinetic energy, KE_final, can be calculated using the formula:
KE_final = 0.5 × m × v^2,
where m is the mass of the proton and v is the final velocity.
Substituting the values of m, u, and v into the formulas will allow you to calculate the increase in kinetic energy, ΔKE.
So, by following these steps, you can find the speed and increase in kinetic energy of the proton.