An athlete executing a long jump leaves the ground at a 27.7∘ angle and travels 7.60m.If this speed were increased by just 5.0%, how much longer would the jump be?

Vi = v sin 27.7

0 = Vi - 9.8 t

t = v sin 27.7 /9.8 that is time to the top
total time in air = 2 t
= 2 v sin 27.7/9.8

u = v cos 27.7
d = (v cos 27.7)(2 v sin 27.7 /9.8)
d = (2 v^2/9.8) cos 27.7 sin 27.7
but 2 cosA sinA = sin 2A
d = (v^2/9.8) sin (55.4)
either do d for v = 7.6 and v=1.05(7.6)
or take derivative
d d/dv = (2v sin 55.4)/9.8
d d = ( 2 v sin 55.4 /9.8) dv

To find out how much longer the jump would be if the speed increased by 5%, we need to calculate the new distance traveled.

First, let's find the initial vertical and horizontal components of the jump.

The vertical displacement (distance) can be found using the formula:
d_vertical = v_initial * t + (1/2) * a * t^2, where
- v_initial is the initial vertical velocity,
- t is the time in seconds, and
- a is the acceleration due to gravity (approximately 9.8 m/s^2).

Since the athlete leaves the ground at an angle of 27.7 degrees, the initial vertical velocity can be found using the formula:
v_initial = v_launch * sin(theta), where
- v_launch is the launch speed, and
- theta is the launch angle (27.7 degrees).

The horizontal displacement (distance) can be found using the formula:
d_horizontal = v_launch * cos(theta) * t.

We can ignore the time component because it will be the same for both the initial and increased speed jumps. So we only need to compare the distances gained horizontally.

To find the new distance traveled:
1. Increase the launch speed (v_launch) by 5% (1.05 times).
2. Calculate the new horizontal distance (d_horizontal_new) using the increased launch speed.

Finally, we can find how much longer the jump would be by subtracting the initial distance from the new distance:
additional distance = d_horizontal_new - d_horizontal_initial.

Let's calculate it step by step:

Step 1: Calculate the initial velocity:

v_initial = v_launch * sin(theta)
= 7.60 m * sin(27.7 degrees)

Step 2: Calculate the initial horizontal distance traveled:

d_horizontal_initial = v_launch * cos(theta) * t
= 7.60 m * cos(27.7 degrees) * t

Step 3: Find the new launch speed (increased by 5%):

v_launch_new = v_launch * 1.05

Step 4: Calculate the new horizontal distance traveled:

d_horizontal_new = v_launch_new * cos(theta) * t

Step 5: Calculate the additional distance:

additional_distance = d_horizontal_new - d_horizontal_initial

Now, you can substitute the given values into the formulas and calculate the additional distance.