Please need your help

Question

A ball is attached to a spring, which is stretched and then let go. The height of the ball is given by the equation y = −1.07 sin (4π/5 t + π/2 ) +1.52
where y is the height above the grounding metre and t is the number of seconds since the ball was released.

(a) At what height was the ball released at? Show your calculation that leads to your answer.

 

(b) What is the maximum height the ball reaches?

 

(c) How many seconds does it take the ball to return to its original position?

 

(d) Draw a sketch of one complete period of the curve, starting at t = 0. Label the maximum and minimum points.

Answer 1

I will help you with (a) - (c); (d) is not practical in this format.

 

y = −1.07 sin (4π/5 t + π/2 ) + 1.52 meters

 

(a)

 

It was released from the height where t = 0.

 

y(0) = −1.07 sin [4π/5 (0) + π/2 ] + 1.52 = -1.07 sin(π/2) + 1.52 = -1.07 + 1.52 = 0.45 meters

 

(b)

 

Maximum height is reached when sin (4π/5 t + π/2 ) = -1

 

y_max = -1.07(-1) + 1.52 = 2.59 meters

 

(c)  

 

It will return after half a period.

 

T = period = 2π/ω

 

ω = 4π/5

 

T/2 = (1/2) [2π / (4π/5)] = 5/4 seconds