The Power of Mathematics: A Ball, a Pebble, and the Force of Gravity
How can we use mathematics to understand the motion of a pebble bouncing off a large ball?
Consider a large ball in the shape of the surface x²/9+y²/4+z²/1=3. A small pebble is thrown so that it hits the ball at the point (3,2,1) and bounces off, normal to the surface, at 30 ft/sec. Find the position vector and velocity vector. Assume that after bouncing off the ball the pebble is acted only by gravity.
Understanding the Dynamics of the Pebble and the Ball
When the pebble hits the large ball at the point (3, 2, 1), it bounces off the surface normal to the ball. The given equation x²/9 + y²/4 + z²/1 = 3 represents the surface of the ball. By differentiating this equation with respect to each variable (x, y, and z), we can find the normal vector to the surface at the point of contact.
Taking the partial derivatives of the equation, we have:
∂/∂x (x²/9 + y²/4 + z²/1) = (2x)/9
∂/∂y (x²/9 + y²/4 + z²/1) = (2y)/4
∂/∂z (x²/9 + y²/4 + z²/1) = (2z)/1
Evaluating the derivatives at the point of contact (3, 2, 1), we get:
∂/∂x (x²/9 + y²/4 + z²/1) = (2 * 3)/9 = 2/3
∂/∂y (x²/9 + y²/4 + z²/1) = (2 * 2)/4 = 1
∂/∂z (x²/9 + y²/4 + z²/1) = (2 * 1)/1 = 2
Hence, the normal vector to the surface at the point of contact is N = ⟨2/3, 1, 2⟩. The velocity vector of the pebble after bouncing off the ball is given as 30 ft/sec, directed opposite to the normal vector. Therefore, the velocity vector is V = -30N = ⟨-(20/3), -30, -60⟩ ft/sec.
Since the pebble is only acted upon by gravity after bouncing off the ball, its position vector remains the same as the point of contact, which is given as P = ⟨3, 2, 1⟩ ft.
The Beauty of Mathematical Calculation
Mathematics allows us to delve into the intricate details of physical phenomena, such as the motion of a pebble bouncing off a large ball. By understanding the mathematical equations that govern such interactions, we can unravel the mysteries of movement and forces at play.
In the scenario provided, we witness the precise calculation of position vectors, velocity vectors, and normal vectors. These vectors not only describe the motion of the pebble but also reveal the underlying principles of physics and mathematics.
The differentiation of the surface equation with respect to each variable showcases the elegance of mathematical operations in determining the normal vector. This normal vector, in turn, influences the direction and magnitude of the pebble's velocity after bouncing off the ball.
Moreover, the simplicity of the pebble's motion post-bounce, influenced solely by gravity, highlights the fundamental force that governs celestial bodies and earthly objects alike. Through mathematical analysis, we can predict and understand the trajectory of the pebble with precision and accuracy.
By exploring such mathematical scenarios, we not only sharpen our analytical skills but also gain a deeper appreciation for the harmonious relationship between mathematics and the physical world. The power of mathematics lies not only in its ability to compute but also in its capacity to reveal the beauty and complexity of the universe.