. How much time will it take a ball to fall to the ground if it rolls off a 25.0° incline on top
of a 3.5 m-high shelf with a speed of 30 m/s?
a. 0.18 s
b. 0.25 s
c. 0.34 s
d. 0.46 s
To solve this problem, we can break it down into two parts: the time it takes for the ball to roll off the incline and the time it takes for the ball to fall from the height of the shelf.
First, calculate the time it takes for the ball to roll off the incline using the following formula:
t = d / v
where:
t = time
d = distance along the incline (hypotenuse of triangle)
v = velocity along the incline
Since the incline is at an angle of 25.0° and the shelf is 3.5 m high, the distance along the incline can be calculated using trigonometry:
d = 3.5 / sin(25.0°)
Next, use the velocity along the incline (which is the horizontal component of the initial velocity) to calculate the time it takes to roll off the incline.
v = 30 * cos(25.0°)
Now calculate the time it takes for the ball to roll off the incline.
t1 = d / v
Next, calculate the time it takes for the ball to fall from the height of the shelf using the following formula:
t2 = sqrt(2 * h / g)
where:
h = height of the shelf (3.5 m)
g = acceleration due to gravity (9.81 m/s^2)
Now, add the times calculated for rolling off the incline and falling from the shelf to get the total time:
total time = t1 + t2
Calculate the total time and check which option it is closest to.