Question

Consider an object on a spring whose position is given by x = (5.00 cm) cos(9.90 s).
(a) What is the maximum speed of the object? (b) When does this maximum speed first occur
after t = 07 (c) What is the maximum of the acceleration of the object? (d) When does the
maximum of the magnitude of the acceleration first occur after t = 0?​

Answer 1

Answer:

Given, displacement x=2×10

−2

cosπt

The magnitude of velocity , v=∣

dt

dx

∣=2π×10

−2

sinπt

So, the velocity will be maximum when sinπt=1 where πt=

2

π

,

2

3π

,..

So for first maximum, πt=

2

π

or t=0.5s

if correct so please rate my answer..

Answer 2

(a) Maximum speed of the object = 49.5 cm/s

(b) Maximum speed first occur after $t=\frac{3\pi}{19.8}$ seconds

(c) Maximum acceleration of the object = 490.05 cm/s

(d) The maximum magnitude of acceleration first occur at $t=\frac{\pi}{9.9}$ seconds

Explanation:

Given, An object on a spring whose position is given by

$x=(5.00\text{cm})\cos(9.90 s^{-1}t)$

(a) In order to find the speed (v) of the object we need to find the first derivative of the given position

Thus,

$v=\frac{dx}{dt}$

$\implies v=\frac{d}{dt}[(5\cos(9.9t))]$

$\implies v=5[-9.9\sin(9.9t)]$

$\implies v=-49.5\sin 9.9t$

The maximum value of the speed will occur when $\sin 9.9t=-1$

Which will occur when $t=\frac{3\pi}{2\times 9.9}$

Thus, at  $t=\frac{3\pi}{19.8}$

Maximum speed

$v_{max}=49.5$ cm/s

(b) The maximum speed first occur after $t=\frac{3\pi}{19.8}$ seconds

(c) Acceleration will be the derivative of velocity

Thus,

$a=\frac{dv}{dt}$

$a=\frac{d}{dt}(-49.5\sin 9.9t$

$\implies a=-49.5\times 9.9\cos 9.9t$

$\implies a=-490.05\cos 9.9t$

Maximum value of acceleration occurs when $\cos 9.9t=-1$

This will happen at $t=\frac{\pi}{9.9}$

Maximum value of acceleration

$a_{max}=490.05$ cm/s²

(d) The maximum magnitude of acceleration first occur after t = 0 at $t=\frac{\pi}{9.9}$ seconds

Hope this answer is helpful.

Know More:

Q: Q. The position of an object moving along x-axis is given by x= a + bt2 where a=8.5m, b=2.5m/s2 and t is measured in seconds. What is its velocity at t=0s and t=2.0s. What is the average velocity b/w t=2.0s and t= 4.0s?

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