A gymnast is performing a routine on the trampoline. At the beginning of each jump, she leaves the trampoline with a vertical velocity of 7.4 m/s upward. What is her vertical acceleration when she is at the apex of each of her jumps?
0m/s^2
0 m/s^2
To find the vertical acceleration at the apex of each jump, we need to consider the motion of the gymnast.
At the apex of the jump, the gymnast's velocity is momentarily zero. This means the gymnast is at the highest point of the jump, and the acceleration due to gravity is acting in the opposite direction.
Given that the initial vertical velocity is 7.4 m/s upward, we can assume the vertical velocity at the apex is zero. Using the equation:
Vf = Vi + at
where:
Vf is the final velocity (zero in this case)
Vi is the initial velocity (7.4 m/s upward)
a is the acceleration
t is the time taken to reach the apex (unknown)
Rearranging the equation to solve for acceleration, we have:
a = (Vf - Vi) / t
Since Vf is zero, the equation becomes:
a = -Vi / t
To find the acceleration, we need to determine the time taken to reach the apex. In this case, it will be half the total time of the jump, as the trajectory is symmetrical.
Let's assume the total time of the jump is t_total. The time to reach the apex would be t_apex = t_total / 2.
Now we can substitute the values into the equation to find the vertical acceleration at the apex:
a = -7.4 m/s / (t_total / 2)
The vertical acceleration at the apex of each jump is equal to -7.4 m/s divided by half of the total time of the jump.
To determine the vertical acceleration of the gymnast at the apex of her jumps, we need to consider the forces acting on her at that point. At the apex, the gymnast reaches the highest point in her jump, where her vertical velocity momentarily becomes zero.
The vertical acceleration can be determined using the following kinematic equation:
v² = u² + 2as
where:
v = final velocity (which is zero at the apex)
u = initial velocity (7.4 m/s upward)
a = acceleration (to be determined)
s = displacement (which is also zero at the apex)
Rearranging the equation, we have:
0 = (7.4 m/s)² + 2a * 0
Since the displacement is zero, the second term on the right side of the equation also becomes zero. Hence, we are left with:
0 = (7.4 m/s)²
Simplifying further, we find that:
0 = 54.76 m²/s²
The equation yields an invalid result, indicating that there is no vertical acceleration at the apex of the gymnast's jumps. This is because at the highest point of the jump, the gymnast is momentarily motionless in the vertical direction, meaning her velocity is zero. Therefore, her vertical acceleration is zero at the apex of each jump.