during a water show the dolphin made a vertical jump of 2.5 m. Determine:
a. The initial velocity they need to make that jump.
b. the time it takes the dolphin to reach the maximum height.
The total time the dolphin remains in the air.
To determine the initial velocity required for the dolphin to make a vertical jump of 2.5 m, we can use the equations of motion and the principle of conservation of energy. Let's assume that the only force acting on the dolphin during the jump is gravity.
a. The initial velocity can be calculated using the formula:
v^2 = u^2 + 2as
where v is the final velocity (which will be zero at the maximum height), u is the initial velocity, a is the acceleration due to gravity (approximately 9.8 m/s^2), and s is the displacement (2.5 m).
Since v = 0, we have:
0 = u^2 + 2(-9.8)(2.5)
0 = u^2 - 49
To solve for u, we can take the square root of both sides:
u^2 = 49
u = ± √49
u = ± 7 m/s
Since velocity is a vector quantity, we take the positive value for the initial velocity:
u = 7 m/s
Therefore, the initial velocity the dolphin needs to make that jump is 7 m/s.
b. The time it takes the dolphin to reach the maximum height can be calculated using the formula:
v = u + at
where v is the final velocity, u is the initial velocity, a is the acceleration due to gravity, and t is the time.
At the maximum height, the final velocity is zero, so we have:
0 = 7 - 9.8t
9.8t = 7
t = 7 / 9.8
t ≈ 0.71 s
Therefore, it takes approximately 0.71 seconds for the dolphin to reach the maximum height.
c. The total time the dolphin remains in the air can be calculated by doubling the time it takes to reach the maximum height, since the time going up is the same as the time coming down:
Total time = 2t
Total time = 2 * 0.71
Total time ≈ 1.42 s
Therefore, the dolphin remains in the air for approximately 1.42 seconds.
To determine the answers to the given questions, we can use the principles of projectile motion. In this case, the dolphin's vertical jump can be considered as a projectile.
a. To find the initial velocity, we can use the formula for vertical displacement:
\( s_y = u_y \cdot t + \frac{1}{2} a_y \cdot t^2 \)
Here, \( s_y \) is the vertical displacement (2.5 m), \( u_y \) is the initial vertical velocity (which we need to find), \( a_y \) is the vertical acceleration (which is the acceleration due to gravity, approximately -9.8 m/s^2), and \( t \) is the time taken for the dolphin to reach the maximum height (which we will find in the next part).
We can rearrange the formula to solve for \( u_y \):
\( u_y = \frac{s_y - \frac{1}{2} a_y \cdot t^2}{t} \)
b. To find the time it takes the dolphin to reach the maximum height, we can use the formula for vertical velocity:
\( v_y = u_y + a_y \cdot t \)
At the maximum height, the vertical velocity becomes zero. Therefore, we can set \( v_y \) to zero and solve for \( t \):
\( 0 = u_y + a_y \cdot t \)
\( t = \frac{-u_y}{a_y} \)
c. The total time the dolphin remains in the air is the double of the time taken to reach the maximum height.
Now, let's substitute the given values and solve for the unknowns:
a. The initial velocity:
\( s_y = 2.5 \, \text{m} \) (vertical displacement)
\( a_y = -9.8 \, \text{m/s}^2 \) (vertical acceleration)
Substituting these values into the formula:
\( u_y = \frac{2.5 \, \text{m} - \frac{1}{2} \cdot (-9.8 \, \text{m/s}^2) \cdot t^2}{t} \)
b. The time taken to reach the maximum height:
\( a_y = -9.8 \, \text{m/s}^2 \) (vertical acceleration)
\( t = \frac{-u_y}{a_y} \)
c. The total time the dolphin remains in the air:
\( t_{\text{total}} = 2 \cdot t_{\text{max height}} \)
By substituting the values for \( t_{\text{max height}} \) into the equation for \( t_{\text{total}} \), we can find the answer.