Question

A boy weighing 50 kg finished a long jump at a distance of 8 m. Considering that he moved along a parabolic path and his angle of jump is \( 45^\circ \), his initial kinetic energy is:

• A. \(960 \text{ J} \)
• B. \(1560 \text{ J} \)
• C. \(2460 \text{ J} \)
• D. \(1960 \text{ J} \)

Hint:

For projectile motion, the range formula helps determine initial velocity, which can then be used to find kinetic energy.

Answer 1

The Correct Option is D

Step 1: Utilize the Projectile Range Formula The range \( R \) of projectile motion is given by: \[ R = \frac{u^2 \sin 2\theta}{g}. \] Substituting \( R = 8 \), \( \theta = 45^\circ \), and \( g = 9.8 \): \[ 8 = \frac{u^2 \sin 90^\circ}{9.8}. \] Since \( \sin 90^\circ = 1 \), solving for \( u^2 \): \[ u^2 = 8 \times 9.8 = 78.4. \]
Step 2: Compute Kinetic Energy The kinetic energy is given by: \[ KE = \frac{1}{2} m u^2. \] Substituting the known values: \[ KE = \frac{1}{2} \times 50 \times 78.4. \] \[ KE = 25 \times 78.4. \] \[ KE = 1960 \text{ J}. \] % Final Answer Thus, the correct answer is option (4): \( 1960 \) J.

Answer 2

Step 1: Apply the Projectile Range Formula The range \( R \) of a projectile launched at an angle \( \theta \) with initial speed \( u \) is given by: \[ R = \frac{u^2 \sin 2\theta}{g}. \] Given \( R = 8 \) m, \( \theta = 45^\circ \), and \( g = 9.8 \, \text{m/s}^2 \), substitute these values: \[ 8 = \frac{u^2 \sin 90^\circ}{9.8}. \] Since \( \sin 90^\circ = 1 \), this simplifies to: \[ u^2 = 8 \times 9.8 = 78.4. \]
Step 2: Calculate the Kinetic Energy The kinetic energy of the projectile is given by: \[ KE = \frac{1}{2} m u^2. \] Substituting mass \( m = 50 \, \text{kg} \) and \( u^2 = 78.4 \): \[ KE = \frac{1}{2} \times 50 \times 78.4 = 25 \times 78.4 = 1960 \text{ J}. \]
Final Answer: Hence, the correct answer is option (4): \[ \boxed{1960 \text{ J}}. \]