An athlete executing a long jump leaves the ground at a 28.0 angle and lands 7.60m away.
part a)What was the takeoff speed?
part b) If this speed were increased by just 8.0 % , how much longer would the jump be?
i got for part a)9.48 m/s
but i need help with part b
L = v^2•sin2α/g ,
v = sqrt(L•g/sin2α) = 9.48 m/s.
New velocity
V =1.08•v = 1.08•9.48= 10.24 m/sþ
New range is L1 = V^2•sin2α/g = 8.87 m.
ΔL = 8.87 – 7.6 = 1.27 m
To solve part b of the problem, we need to first calculate the original jump distance and then calculate the new jump distance after increasing the takeoff speed by 8%.
Let's start with the original jump distance. We know that the athlete landed 7.60m away from the starting point.
The horizontal distance traveled in a projectile motion (such as a long jump) can be calculated using the formula:
Horizontal Distance = Initial Velocity * Cosine(theta) * Time
In this case, the time is the time the athlete is in the air, which can be calculated using the vertical motion formula:
Time = (2 * Initial Velocity * Sine(theta)) / g
Where g is the acceleration due to gravity (approximately 9.8 m/s^2) and theta is the takeoff angle (28.0 degrees).
Simplifying the equations and plugging in the known values:
Time = (2 * Initial Velocity * Sine(28.0)) / 9.8
Plugging the obtained value of Time into the first equation:
7.60m = Initial Velocity * Cosine(28.0) * [(2 * Initial Velocity * Sine(28.0)) / 9.8]
Simplifying this equation will help us find the initial velocity.
Now, solving this equation for Initial Velocity will give us the value for part a, which you have already found to be 9.48 m/s.
To find the new jump distance after increasing the takeoff speed by 8%, we can calculate the new initial velocity:
New Initial Velocity = 1.08 * Initial Velocity (since it increased by 8%)
Plugging this new value into the horizontal distance formula, with the same takeoff angle and calculating it with the same formula:
New Distance = New Initial Velocity * Cosine(theta) * Time
Here, Time remains the same as calculated in part a, and we can substitute the new value of New Initial Velocity and solve for New Distance.
New Distance = (1.08 * Initial Velocity) * Cosine(28.0) * Time
By calculating this equation, we can find how much longer the jump would be if the takeoff speed increased by 8%.