A juggler traveling in a train on level track throws a ball straight up, relative to the train, with a speed of 5.75 m/s. The train has a velocity of 19 m/s due east.
(a) As observed by the juggler, what is the ball's total time of flight?
s
What is the displacement of the ball during its rise?
m
To answer the first question, we need to consider the motion of the ball from the perspective of the juggler. Since the juggler is also moving with the train, we need to add the velocities of the train and the ball to find the total velocity of the ball relative to an observer on the ground.
The velocity of the train is given as 19 m/s due east. The ball is thrown straight up, so its velocity component in the eastward direction is 19 m/s. However, the ball is thrown upward, so its vertical velocity component is 5.75 m/s.
To find the total velocity of the ball relative to the ground observer, we can use the Pythagorean theorem. The total velocity is given by the square root of the sum of the squares of the horizontal and vertical components.
Total velocity = sqrt((19 m/s)^2 + (5.75 m/s)^2)
Once we have the total velocity of the ball relative to the ground observer, we can calculate the total time of flight. Since the ball is thrown straight up, it will reach its highest point when its vertical velocity becomes zero. From there, it will fall back down and land on the ground.
The time taken to reach the highest point is given by dividing the initial vertical velocity by the acceleration due to gravity. Taking into account that the acceleration due to gravity is approximately 9.8 m/s^2 and the initial vertical velocity is 5.75 m/s, we can calculate the time to reach the highest point.
Time to reach highest point = (initial vertical velocity) / (acceleration due to gravity)
Once the ball reaches the highest point, it will start falling back down. The time taken for the ball to fall back down is the same as the time taken to reach the highest point.
Finally, to find the total time of flight, we can simply double the time taken to reach the highest point.
Total time of flight = 2 * (time to reach highest point)