An athlete executing a long jump leaves the ground at a 32.3 angle and travels 7.84 m. What was the takeoff speed?

If this speed were increased by just 5.0%, how much longer would the jump be?

Neglecting air friction? Right.

vertical component speed=Vsin32.2
time in air..
hf=hi+vv*t-4.9t^2
0=V *sin32.2 t-4.9t^2
t=V *sin32.2/4.9

now, knowing the time in air, distance traveled is
7.84 =V*Cos32.2 * t solve for V given t above.

To find the takeoff speed of the athlete executing a long jump, we can use the principles of projectile motion. Here's how you can calculate the takeoff speed:

1. Start by breaking down the initial velocity into its horizontal and vertical components. Since the athlete leaves the ground at an angle of 32.3°, we can use trigonometry to find the vertical and horizontal components.

Vertical component: V_y = V * sin(θ)
Horizontal component: V_x = V * cos(θ)

In this case, the angle (θ) is 32.3°, and we need to find V.

2. Knowing the horizontal displacement, we can use the horizontal component of velocity (V_x) and the time of flight to find the total time spent in the air.

Horizontal displacement (x) = V_x * t

In this case, the horizontal displacement is given as 7.84 m.

3. Next, let's find the time of flight (t) using the vertical component of velocity (V_y) and the acceleration due to gravity (g). Since the motion only occurs vertically, we can use the equation:

Vertical displacement (y) = V_y * t + (1/2) * g * t^2

In this case, the vertical displacement is zero since the takeoff and landing points are at the same height.

4. The total time of flight (t) can now be found by rearranging the equation above to solve for t:

0 = V_y * t + (1/2) * g * t^2

Simplifying the equation gives us a quadratic equation which can be solved to find t.

Once you've found the time of flight, you can substitute it back into the equation for horizontal displacement (x) and solve for V_x.

5. After finding both V_x and V_y, you can combine them using the Pythagorean theorem to find the takeoff speed (V):

V = sqrt(V_x^2 + V_y^2)

Now, to find how much longer the jump would be if the speed were increased by 5.0%, you can follow these steps:

1. Calculate the new takeoff speed by increasing the original takeoff speed by 5.0%.

New takeoff speed = (1 + 0.05) * original takeoff speed

2. Use the new takeoff speed to calculate the new horizontal displacement.

New horizontal displacement = new V_x * t

3. Compare the new horizontal displacement to the original horizontal displacement to find the difference. This will give you the additional distance covered due to the increased speed.

Additional distance = new horizontal displacement - original horizontal displacement

By following these steps, you would be able to calculate the takeoff speed and determine the additional distance covered if the speed were increased by 5.0%.

9.41m/s