The kinetic energy of a proton is half its internal energy. What is the proton's speed?
Kinetic energy is (Yu - 1) (mp) (c)^2
= 1/2 (mp) (c)^2
Yu = 1.5
Yu = 1/(square root) 1 - (u/c)^2
1/(square root) 1 - (u/c)^2 = 1.5
u= 0.745c
How do I solve for u? I'm extremely confused on how to do this. I know you square 1/(square root) 1 - (u/c)^2 to both sides esp. 1.5 to get rid of the square root. But up to there I'm confused to how to solve for u
1/√(1-(u/c)^2) = 3/2
√(1-(u/c)^2) = 2/3
(1-(u/c)^2) = 4/9
(u/c)^2 = 5/9
u/c = √5/3 = .745
u = .745c
thank you
To solve for "u" in the equation 1/(√(1 - (u/c)^2)) = 1.5, you need to follow these steps:
Step 1: Square both sides of the equation to eliminate the square root:
(1/(√(1 - (u/c)^2)))^2 = (1.5)^2
Step 2: Simplify the left side of the equation:
1/(1 - (u/c)^2) = 2.25
Step 3: Cross multiply to eliminate the denominator:
1 = 2.25 - 2.25(u/c)^2
Step 4: Rearrange the equation to isolate (u/c)^2:
2.25(u/c)^2 = 2.25 - 1
Step 5: Simplify the right side of the equation:
2.25(u/c)^2 = 1.25
Step 6: Divide both sides of the equation by 2.25 to solve for (u/c)^2:
(u/c)^2 = 1.25 / 2.25
Step 7: Take the square root of both sides to determine (u/c):
u/c = √(1.25 / 2.25)
Step 8: Multiply both sides of the equation by c to solve for u:
u = √(1.25 / 2.25) * c
Now, you can substitute the value of "c" (speed of light in a vacuum) into the equation to calculate the proton's speed "u".
To solve for the proton's speed, u, in the equation 1/(√(1 - (u/c)^2)) = 1.5, you can follow these steps:
Step 1: Square both sides of the equation to eliminate the square root:
(1/(√(1 - (u/c)^2)))^2 = (1.5)^2
Step 2: Simplify the left-hand side:
1/(1 - (u/c)^2) = 2.25
Step 3: Multiply both sides of the equation by (1 - (u/c)^2):
1 = 2.25 - 2.25(u/c)^2
Step 4: Rearrange the equation and combine like terms:
2.25(u/c)^2 = 2.25 - 1
Step 5: Simplify the right-hand side:
2.25(u/c)^2 = 1.25
Step 6: Divide both sides of the equation by 2.25:
(u/c)^2 = 1.25 / 2.25
Step 7: Take the square root of both sides:
u/c = √(1.25 / 2.25)
Step 8: Multiply both sides by c:
u = c * √(1.25 / 2.25)
Now you can substitute the appropriate value for the speed of light, c, and calculate the value of u.