Final answer:
The range of a projectile, such as a long jumper, is calculated using the initial velocity components and the projectile motion formula. For a jump with a velocity of 10 m/s at a 30-degree angle, one must find the horizontal (vx) and vertical (vy) velocities, the time to reach the peak (t), and then calculate the range (R) by multiplying 2t with vx.
Step-by-step explanation:
The student is asking how to calculate the range, or horizontal distance, of a long jumper who leaps with a velocity of 10 m/s at an angle of 30 degrees from the horizontal. To find the range of a projectile, we use the formula:
R = (v² sin 2θ) / g
Where:
R = Range (horizontal distance)
v = Initial velocity
θ = Angle of projection
g = Acceleration due to gravity (9.81 m/s² on Earth)
However, here we must also consider the initial velocity components in both the x (horizontal) and y (vertical) directions. The horizontal component is given by vx = v cos θ and the vertical component is given by vy = v sin θ. Since the trajectory's peak occurs at half the total time of flight and is symmetric, we can use the time it takes for the jumper to reach the peak (t) to find the total time of flight (2t), then multiply it by the horizontal component of velocity to find the range:
R = vx × 2t
To solve for the time of flight (t), we use the vertical motion equation:
t = (vy) / g
Therefore, by combining all these equations with proper values, we can calculate the range of the jump.