The equation for the displacement in meters of an object moving in SHM is x(t) = 1.50 cos (4.20t) where t is in seconds. (a) What is the maximum speed of the object? (b) At what time does it first reach the maximum speed?
I already got 6.3m/s for A and I got B wrong. I thought it was 1.12 s
v = - 1.50 (4.20) sin 4.2 t
the maximum speed is when the sine function is +1 or -1
since the sign is negative the sine function reaches -1 first.
|v|max = max speed = 1.5*4.2 = 6.3 m/s
this is when 4.2 t is pi/2 radians
t = pi/8.4 = .374 s
To find the maximum speed of the object in Simple Harmonic Motion (SHM), we can differentiate the equation for displacement with respect to time, and then take the absolute value of this derivative. This will give us the magnitude of the velocity function.
(a) The equation for velocity is given by v(t) = dx(t)/dt = -1.50 * 4.20 * sin(4.20t)
To find the maximum speed, we need to find the maximum value of v(t). Since the sine function oscillates between -1 and 1, the maximum value of sin(4.20t) is 1. Therefore, the maximum speed will be:
|v(max)| = |-1.50 * 4.20 * 1| = 6.30 m/s
So, you correctly obtained the maximum speed as 6.3 m/s.
(b) Now, let's find the time at which the object first reaches the maximum speed. The maximum speed occurs when the sine function is equal to 1. Setting sin(4.20t) = 1 and solving for t, we have:
1 = sin(4.20t)
To find the value of t when sin(4.20t) equals 1, we can use the inverse sine function (sin^(-1)) or arcsin. Taking the inverse sine of both sides, we get:
4.20t = sin^(-1)(1)
Since sin^(-1)(1) = π/2 radians or 90 degrees, we have:
4.20t = π/2
Solving for t:
t = (π/2) / 4.20 ≈ 0.374 s
Therefore, the object first reaches the maximum speed at approximately 0.374 seconds.
It appears that your answer for part (b) was incorrect. The correct time at which the object first reaches the maximum speed is approximately 0.374 seconds.