A ball is thrown vertically upward, starting at y = 0, with a velocity of 15.0 m/s.
(a) Calculate the maximum vertical displacement of the ball.
(b) Calculate the time required for the ball to reach its highest point.
(c) What is the ball's acceleration when it is at its highest point?
(d) How long does the ball take to reach the point y = 0 after it reaches its highest point?
(e) What is its velocity when it returns to the level from which it started (y = 0)?
Is there no part of this you can figure out?
To answer these questions, we can use the equations of motion relating displacement, velocity, acceleration, and time. The key equation for vertical motion is:
y = y0 + v0*t + (1/2)*a*t^2
Where:
y = vertical displacement
y0 = initial height (in this case, y = 0)
v0 = initial velocity (upward = positive)
t = time
a = acceleration (in this case, due to gravity = -9.8 m/s^2, downward = negative)
Now let's solve each part of the problem step by step:
(a) To calculate the maximum vertical displacement, we need to find the time at which the ball reaches its highest point. At the highest point of a vertical trajectory, the vertical velocity becomes zero. Therefore, we can use the equation:
v = v0 + a*t
At its highest point, v = 0:
0 = 15.0 - 9.8*t
Solving for t:
9.8*t = 15.0
t = 15.0 / 9.8 ≈ 1.53 seconds
Now substitute this time into the equation for vertical displacement:
y = y0 + v0*t + (1/2)*a*t^2
y = 0 + 15.0*1.53 + (1/2)*(-9.8)*(1.53)^2
y ≈ 11.31 meters
Therefore, the maximum vertical displacement of the ball is approximately 11.31 meters.
(b) The time required for the ball to reach its highest point is the same time we found in part (a), which is approximately 1.53 seconds.
(c) At the highest point, the ball's velocity is zero, so the acceleration is also zero. Therefore, the ball's acceleration when it is at its highest point is 0 m/s^2.
(d) To find the time it takes for the ball to reach y = 0 after reaching its highest point, we can use the equation:
y = y0 + v0*t + (1/2)*a*t^2
Setting y = 0 and solving for t:
0 = 11.31 + 0 - (1/2)*9.8*t^2
9.8*t^2 = 11.31
t^2 = 11.31 / 9.8
t ≈ √(11.31 / 9.8)
t ≈ √1.153
t ≈ 1.074 seconds
Therefore, it takes approximately 1.074 seconds for the ball to reach y = 0 after reaching its highest point.
(e) Finally, to find the velocity of the ball when it returns to the level from which it started (y = 0), we can use the equation:
v = v0 + a*t
Using the time we found in part (d), t ≈ 1.074 seconds:
v = 15.0 - 9.8*1.074
v ≈ 15.0 - 10.518
v ≈ 4.482 m/s
Therefore, the velocity of the ball when it returns to y = 0 is approximately 4.482 m/s.