A ball can be hit faster if it is projected without spin but it can be hit farther if it is projected with backspin. Measurements are presented in this paper of the tradeoff between speed. Spin for a baseball impacting a baseball bat. The results are inconsistent with a collision model in which the ball rolls off the bat and instead imply tangential compliance in the ball, the bat, or both. If the results are extrapolated to the higher speeds that are typical of the game of baseball, they suggest that a curveball can be hit with greater backspin than a fastball, but by an amount that is less than would be the case in the absence of tangential compliance.

Particle scattering experiments have been conducted for many years to probe the structure of the atom, the atomic nucleus, and the nucleons. By comparison, very few scattering experiments have been conducted with macroscopic objects. In this paper we describe an experiment on the scattering of a baseball by a baseball bat, not to probe the structure of the bat but to determine how the speed and spin of the outgoing ball depends on the scattering angle. In principle the results could be used to determine an appropriate force law for the interaction, but we focus attention in this paper on directly observable parameters. The main purpose of the experiment was to determine the amount of backspin that can be imparted to a baseball by striking it at a point below the center of the ball. The results are of a preliminary nature in that they were obtained at lower ball speeds than those encountered in the field. As such, the experiment could easily be demonstrated in the classroom or repeated in an undergraduate laboratory as an introduction to scattering problems in general.

A golf ball is normally lofted with backspin so that the aerodynamic lift force will carry the ball as far as possible. For the same reason, a baseball will also travel farther if it is struck with backspin. It also travels farther if it is launched at a higher speed. In general there is a tradeoff between the spin and speed that can be imparted to a ball, which is affected in baseball by the spin and speed of the pitched ball. Sawicki, et al.[1], henceforth referred to as SHS, examined this problem and concluded that a curveball can be batted farther than a fastball despite the higher incoming and outgoing speed of the fastball. The explanation is that a curveball is incident with topspin. Hence the ball is already spinning in the correct direction to exit with backspin. A fastball is incident with backspin so the spin direction needs to be reversed in order to exit with backspin. As a result, the magnitude of the backspin imparted to a curveball is larger than that imparted to a fastball for any given bat speed and impact point on the bat, even allowing for the lower incident speed of a curveball. According to SHS,[1] the larger backspin on a hit curveball more than compensates the smaller hit ball speed and travels farther than a fastball, a conclusion that has been challenged in the literature.[2] SHS[1] assumed that a batted ball of radius rݑŸritalic_r will roll off the bat with a spin ωݜ”\omegaitalic_ω given by rω=vxݑŸݜ”subscriptݑ£ݑ¥r\omega=v_xitalic_r italic_ω = italic_v start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT where vxsubscriptݑ£ݑ¥v_xitalic_v start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT is the tangential speed of the ball as it exits the bat. However, a number of recent experiments[3, 4, 5, 6, 7] have shown that balls do not roll when they bounce. Rather, a ball incident obliquely on a surface will grip during the bounce and usually bounces with rω>vxݑŸݜ”subscriptݑ£ݑ¥r\omega>v_xitalic_r italic_ω >italic_v start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT if the angle of incidence is within about 45∘superscript4545^\circ45 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT to the normal. The actual spin depends on the tangential compliance or elasticity of the mating surfaces and is not easy to calculate accurately. For that reason we present in this paper measurements of speed, spin and rebound angle of a baseball impacting with a baseball bat. The implications for batted ball speed. Spin are also described.

II Experimental procedures

A baseball was dropped vertically onto a stationary, hand-held baseball bat to determine the rebound speed and spin as functions of (a) the scattering angle and (b) the magnitude and direction of spin of the incident ball. The impact distance from the longitudinal axis of the bat was varied on a random basis in order to observe scattering at angles up to about 120∘superscript120120^\circ120 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT away from the vertical. Measurements were made by filming each bounce with a video camera operating at 100 frames/s, although satisfactory results were also obtained at 25 frames/s. The bat chosen for the present experiment was a modified Louisville Slugger model R161 wood bat of length 84 cm (33 in.) with a barrel diameter of 6.67 cm (2 5858\frac58divide start_ARG 5 end_ARG start_ARG 8 end_ARG in.) and mass M=0.989ݑ€0.989M=0.989italic_M = 0.989 kg (35 oz). The center of mass of the bat was located 26.5 cm from the barrel end of the bat. The moments of inertia about axes through the center of mass and perpendicular and parallel, respectively, to the longitudinal axis of the bat were 0.0460 and 4.39 x 10-44{}^-4start_FLOATSUPERSCRIPT - 4 end_FLOATSUPERSCRIPT kg-m22{}^2start_FLOATSUPERSCRIPT 2 end_FLOATSUPERSCRIPT. The ball was a Wilson A1010, having a mass 0.145 kg and diameter 7.2 cm. The bat was held in a horizontal position by one hand. The ball was dropped from a height of about 0.8 m using the other hand.8 m using the other hand. A plumb bob was used to establish a true vertical in the video image and to help align both the bat and the ball. The ball was dropped either with or without spin. In order to spin the ball, a strip of felt was wrapped around a circumference and the ball was allowed to fall vertically while holding the top end of the felt strip. A line drawn around a circumference was used to determine the ball orientation in each frame in order to measure its spin. The impact distance along the axis was determined by eye against marks on the barrel to within about 5 mm. If the ball landed 140 to 160 mm from the barrel end of the bat the bounce was accepted. Bounces outside this range were not analyzed.

The velocity of the ball immediately prior to and after impact was determined to within 2% by extrapolating data from at least three video frames before and after each impact. The horizontal velocity was obtained from linear fits to the horizontal coordinates of the ball and the vertical velocity was obtained from quadratic fits assuming a vertical acceleration of 9.8 m/s22{}^2start_FLOATSUPERSCRIPT 2 end_FLOATSUPERSCRIPT. Additional measurements were made by bouncing the ball on a hard wood floor to determine the normal and tangential COR, the latter defined below in Eq. 1, and a lower limit on the coefficient of sliding friction (μksubscriptݜ‡ݑ˜\mu_kitalic_μ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT) between the ball and the floor. The COR values were determined by dropping the ball with and without spin from a height of about 1.5 m to impact the floor at a speed of 5.6±0.3plus-or-minus5.60.35.6\pm 0.3{}5.6 ± 0.3m/s. The incident ball spin was either zero, -72±2plus-or-minus722-72\pm 2- 72 ± 2 rad/s or +68±3plus-or-minus683+68\pm 3+ 68 ± 3 rad/s. The normal COR eysubscriptݑ’ݑ¦e_yitalic_e start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT was 0.59±0.01plus-or-minus0.590.010.59\pm 0.010.59 ± 0.01, and the tangential COR exsubscriptݑ’ݑ¥e_xitalic_e start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT was 0.17±0.03plus-or-minus0.170.030.17\pm 0.030.17 ± 0.03, corresponding to a rebound spin ω2≈0.16ω1subscriptݜ”20.16subscriptݜ”1\omega_2\approx 0.16\omega_1italic_ω start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ≈ 0.16 italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, where ω1subscriptݜ”1\omega_1italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT is the incident spin. If a spinning baseball is dropped vertically onto a hard floor, then it would bounce with ω2=0.29ω1subscriptݜ”20.29subscriptݜ”1\omega_2=0.29\omega_1italic_ω start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 0.29 italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT if ex=0subscriptݑ’ݑ¥0e_x=0italic_e start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT = 0 (as assumed by SHS[1]). The lower limit on μksubscriptݜ‡ݑ˜\mu_kitalic_μ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT was determined by throwing the ball obliquely onto the floor at angles of incidence between 25∘superscript2525^\circ25 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT and 44∘superscript4444^\circ44 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT to the horizontal, at speeds from 3.5 to 4.2 m/s and with negligible spin. The value of μksubscriptݜ‡ݑ˜\mu_kitalic_μ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT was found from the data at low angles of incidence to be larger than 0.31±0.02plus-or-minus0.310.020.31\pm 0.020.31 ± 0.02. At angles of incidence between 30∘superscript3030^\circ30 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT and 44∘superscript4444^\circ44 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT the ball did not slide throughout the bounce but gripped the floor during the bounce, with ex=0.14±0.02subscriptݑ’ݑ¥plus-or-minus0.140.02e_x=0.14\pm 0.02italic_e start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT = 0.14 ± 0.02.

III Theoretical bounce models

Consider the situation shown in Fig. R. In a low speed collision the bat. The ball will remain approximately circular in cross section. In a low speed collision the bat. The ball will remain approximately circular in cross section. If the impact parameter is Eݐ¸Eitalic_E, then the line joining the bat and ball centers is inclined at an angle βݛ½\betaitalic_β to the horizontal where cosβ=E/(r+R)ݛ½ݐ¸ݑŸݑ…\,\beta=E/(r+R)italic_β = italic_E / ( italic_r + italic_R ). The ball is incident at an angle θ1=90-βsubscriptݜƒ190ݛ½\theta_1=90-\betaitalic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 90 - italic_β to the line joining the bat and ball centers and will rebound at an angle θ2subscriptݜƒ2\theta_2italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT. The ball is therefore scattered at an angle α=θ1+θ2ݛ¼subscriptݜƒ1subscriptݜƒ2\alpha=\theta_1+\theta_2italic_α = italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT. During the collision, the ball experiences a tangential force Fݐ¹Fitalic_F and a normal force NݑNitalic_N. For the low-speed collisions investigated here, the ball-bat force acts essentially at a point, so that the angular momentum of the ball about that point is conserved. Indeed, low-speed collisions of tennis balls are consistent with angular momentum conservation.[4] However, high-speed collisions of tennis balls are known not to conserve angular momentum.[5] A phenomenological way to account for non-conservation of angular momentum is to assume that the normal force NݑNitalic_N does not act through the center of mass of the ball but is displaced from it by the distance Dݐ·Ditalic_D,[4] as shown in Fig. 1 and discussed more fully below.

The collision is essentially equivalent to one between a ball and a plane surface inclined at angle θ1subscriptݜƒ1\theta_1italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT to the horizontal. Suppose that the ball is incident with angular velocity ω1subscriptݜ”1\omega_1italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and speed v1subscriptݑ£1v_1italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT. Let vy1=v1cosθ1subscriptݑ£ݑ¦1subscriptݑ£1cossubscriptݜƒ1v_y1=v_1\,\rm cos\,\theta_1italic_v start_POSTSUBSCRIPT italic_y 1 end_POSTSUBSCRIPT = italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT roman_cos italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT denote the component of the incident ball speed normal to the surface and vx1=v1sinθ1subscriptݑ£ݑ¥1subscriptݑ£1sinsubscriptݜƒ1v_x1=v_1\,\rm sin\,\theta_1italic_v start_POSTSUBSCRIPT italic_x 1 end_POSTSUBSCRIPT = italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT roman_sin italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT denote the tangential component. The ball will bounce at speed vy2subscriptݑ£ݑ¦2v_y2italic_v start_POSTSUBSCRIPT italic_y 2 end_POSTSUBSCRIPT in a direction normal to the surface, with tangential speed vx2subscriptݑ£ݑ¥2v_x2italic_v start_POSTSUBSCRIPT italic_x 2 end_POSTSUBSCRIPT and angular velocity ω2subscriptݜ”2\omega_2italic_ω start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT. If the bat is initially at rest, it will recoil with velocity components Vysubscriptݑ‰ݑ¦V_yitalic_V start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT and Vxsubscriptݑ‰ݑ¥V_xitalic_V start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT respectively perpendicular and parallel to the surface, where the velocity components refer to the impact point on the bat. The recoil velocity at the handle end or the center of mass of the bat are different since the bat will tend to rotate about an axis near the end of the handle.

The bounce can be characterized in terms of three independent parameters: the normal coefficient of restitution (COR) ey=(vy2-Vy)/vy1subscriptݑ’ݑ¦subscriptݑ£ݑ¦2subscriptݑ‰ݑ¦subscriptݑ£ݑ¦1e_y=(v_y2-V_y)/v_y1italic_e start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT = ( italic_v start_POSTSUBSCRIPT italic_y 2 end_POSTSUBSCRIPT - italic_V start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ) / italic_v start_POSTSUBSCRIPT italic_y 1 end_POSTSUBSCRIPT; the tangential COR, exsubscriptݑ’ݑ¥e_xitalic_e start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT, defined by

ex=-(vx2-rω2-[Vx-RΩ])(vx1-rω1)subscriptݑ’ݑ¥subscriptݑ£ݑ¥2ݑŸsubscriptݜ”2delimited-[]subscriptݑ‰ݑ¥ݑ…Ωsubscriptݑ£ݑ¥1ݑŸsubscriptݜ”1e_x=-\frac(v_x2-r\omega_2-[V_x-R\Omega])(v_x1-r\omega_1)italic_e start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT = - divide start_ARG ( italic_v start_POSTSUBSCRIPT italic_x 2 end_POSTSUBSCRIPT - italic_r italic_ω start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT - [ italic_V start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT - italic_R roman_Ω ] ) end_ARG start_ARG ( italic_v start_POSTSUBSCRIPT italic_x 1 end_POSTSUBSCRIPT - italic_r italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) end_ARG (1)
where ΩΩ\Omegaroman_Ω is the angular velocity of the bat about the longitudinal axis immediately after the collision; and the parameter Dݐ·Ditalic_D. The two coefficients of restitution are defined respectively in terms of the normal and tangential speeds of the impact point on the ball, relative to the bat, immediately after and immediately before the bounce. The bounce can also be characterized in terms of apparent coefficients of restitution, ignoring recoil and rotation of the bat. That is, one can define the apparent normal COR eA=vy2/vy1subscriptݑ’ݐ´subscriptݑ£ݑ¦2subscriptݑ£ݑ¦1e_A=v_y2/v_y1italic_e start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT = italic_v start_POSTSUBSCRIPT italic_y 2 end_POSTSUBSCRIPT / italic_v start_POSTSUBSCRIPT italic_y 1 end_POSTSUBSCRIPT and the apparent tangential COR, eTsubscriptݑ’ݑ‡e_Titalic_e start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT, given by

eT=-(vx2-rω2)(vx1-rω1)subscriptݑ’ݑ‡subscriptݑ£ݑ¥2ݑŸsubscriptݜ”2subscriptݑ£ݑ¥1ݑŸsubscriptݜ”1e_T=-\frac(v_x2-r\omega_2)(v_x1-r\omega_1)italic_e start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT = - divide start_ARG ( italic_v start_POSTSUBSCRIPT italic_x 2 end_POSTSUBSCRIPT - italic_r italic_ω start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) end_ARG start_ARG ( italic_v start_POSTSUBSCRIPT italic_x 1 end_POSTSUBSCRIPT - italic_r italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) end_ARG (2)
There are three advantages of defining apparent COR values in this manner. The first is that apparent COR values are easier to measure since there is no need to measure the bat speed. Angular velocity before or after the collision (provided the bat speed is zero before the collision). The second advantage is that the batted ball speed can be calculated from the measured apparent COR values for any given initial bat speed simply by a change of reference frame. We show how this is done in Appendix B. The third advantage is that the algebraic solutions of the collision equations are considerably simplified and therefore more easily interpreted. Apparent and actual values of the COR are related by the expressions

eA=(ey-ry)(1+ry)subscriptݑ’ݐ´subscriptݑ’ݑ¦subscriptݑŸݑ¦1subscriptݑŸݑ¦e_A=\frac(e_y-r_y)(1+r_y)italic_e start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT = divide start_ARG ( italic_e start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT - italic_r start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ) end_ARG start_ARG ( 1 + italic_r start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ) end_ARG (3)
and

eT=(ex-rx)(1+rx)+52[Dr](rx1+rx)vy1(1+eA)(vx1-rω1),subscriptݑ’ݑ‡subscriptݑ’ݑ¥subscriptݑŸݑ¥1subscriptݑŸݑ¥52delimited-[]ݐ·ݑŸsubscriptݑŸݑ¥1subscriptݑŸݑ¥subscriptݑ£ݑ¦11subscriptݑ’ݐ´subscriptݑ£ݑ¥1ݑŸsubscriptݜ”1e_T=\frac(e_x-r_x)(1+r_x)+\frac52\left[\fracDr\right]\left% (\fracr_x1+r_x\right)\fracv_y1(1+e_A)(v_x1-r\omega_1)\,,italic_e start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT = divide start_ARG ( italic_e start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT - italic_r start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ) end_ARG start_ARG ( 1 + italic_r start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ) end_ARG + divide start_ARG 5 end_ARG start_ARG 2 end_ARG [ divide start_ARG italic_D end_ARG start_ARG italic_r end_ARG ] ( divide start_ARG italic_r start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_ARG start_ARG 1 + italic_r start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_ARG ) divide start_ARG italic_v start_POSTSUBSCRIPT italic_y 1 end_POSTSUBSCRIPT ( 1 + italic_e start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ) end_ARG start_ARG ( italic_v start_POSTSUBSCRIPT italic_x 1 end_POSTSUBSCRIPT - italic_r italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) end_ARG , (4)
where the recoil factors, rysubscriptݑŸݑ¦r_yitalic_r start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT and rxsubscriptݑŸݑ¥r_xitalic_r start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT, are the ratios of effective ball to bat masses for normal and tangential collisions, respectively. An expression for rysubscriptݑŸݑ¦r_yitalic_r start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT was derived by Cross:[8]

ry=m(1M+b2I0)subscriptݑŸݑ¦ݑš1ݑ€superscriptݑ2subscriptݐ¼0r_y\,=\,m\left(\frac1M\,+\fracb^2I_0\right)\,italic_r start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT = italic_m ( divide start_ARG 1 end_ARG start_ARG italic_M end_ARG + divide start_ARG italic_b start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_I start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG ) (5)
and an expression for rxsubscriptݑŸݑ¥r_xitalic_r start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT is derived in Appendix A:

rx=27m(1M+b2I0+R2Iz).subscriptݑŸݑ¥27ݑš1ݑ€superscriptݑ2subscriptݐ¼0superscriptݑ…2subscriptݐ¼ݑ§r_x\,=\,\frac27m\left(\frac1M\,+\fracb^2I_0\,\,+\,\fracR^2% I_z\right)\,.italic_r start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT = divide start_ARG 2 end_ARG start_ARG 7 end_ARG italic_m ( divide start_ARG 1 end_ARG start_ARG italic_M end_ARG + divide start_ARG italic_b start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_I start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG + divide start_ARG italic_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_I start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT end_ARG ) . (6)
In these expressions, mݑšmitalic_m is the ball mass, I0subscriptݐ¼0I_0italic_I start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and Izsubscriptݐ¼ݑ§I_zitalic_I start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT are the moments of inertia (MOI) about an axis through the center of mass and perpendicular and parallel, respectively, to the longitudinal axis of the bat, and bݑbitalic_b is the distance parallel to the longitudinal axis between the impact point and the center of mass. For the bat used in the experiments at an impact distance 15 cm from the barrel end of the bat, rysubscriptݑŸݑ¦r_yitalic_r start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT=0.188 and rxsubscriptݑŸݑ¥r_xitalic_r start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT=0.159, assuming the bat is free at both ends. The exit parameters of the ball are independent of whether the handle end is free or hand-held, as described previously by the authors.[9, 10] Eq. 4 will not be used in this paper except for some comments in Sec. C and for comparison with SHS[1] who assumed in their calculations that ex=0subscriptݑ’ݑ¥0e_x=0italic_e start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT = 0 and D=0ݐ·0D=0italic_D = 0, implying eT=-0.14subscriptݑ’ݑ‡0.14e_T=-0.14italic_e start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT = - 0.14 for our bat. As discussed more fully in the next section, we find better agreement with our data when eT=0subscriptݑ’ݑ‡0e_T=0italic_e start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT = 0.

From the definition of the parameter Dݐ·Ditalic_D, the normal force exerts a torque resulting in a change in angular momentum of the ball about the contact point given by

(Iω2+mrvx2)-(Iω1+mrvx1)=-D∫Nݑ‘t=-mD(1+eA)vy1ݐ¼subscriptݜ”2ݑšݑŸsubscriptݑ£ݑ¥2ݐ¼subscriptݜ”1ݑšݑŸsubscriptݑ£ݑ¥1ݐ·ݑdifferential-dݑ¡ݑšݐ·1subscriptݑ’ݐ´subscriptݑ£ݑ¦1\left(I\omega_2+mrv_x2\right)-\left(I\omega_1+mrv_x1\right)=-D\int Ndt% =-mD(1+e_A)v_y1( italic_I italic_ω start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + italic_m italic_r italic_v start_POSTSUBSCRIPT italic_x 2 end_POSTSUBSCRIPT ) - ( italic_I italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_m italic_r italic_v start_POSTSUBSCRIPT italic_x 1 end_POSTSUBSCRIPT ) = - italic_D ∫ italic_N italic_d italic_t = - italic_m italic_D ( 1 + italic_e start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ) italic_v start_POSTSUBSCRIPT italic_y 1 end_POSTSUBSCRIPT (7)
where I=αmr2ݐ¼ݛ¼ݑšsuperscriptݑŸ2I=\alpha mr^2italic_I = italic_α italic_m italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT is the moment of inertia of the ball about its center of mass. For a solid sphere, α=2/5ݛ¼25\alpha=2/5italic_α = 2 / 5, although Brody has recently shown that α≈0.378ݛ¼0.378\alpha\approx 0.378italic_α ≈ 0.378 for a baseball.[11] Eqs. 2 and 7 can be solved to show that

vx2vx1=(1-αeT)(1+α)+α(1+eT)(1+α)(rω1vx1)-D(1+eA)r(1+α)(vy1vx1)subscriptݑ£ݑ¥2subscriptݑ£ݑ¥11ݛ¼subscriptݑ’ݑ‡1ݛ¼ݛ¼1subscriptݑ’ݑ‡1ݛ¼ݑŸsubscriptݜ”1subscriptݑ£ݑ¥1ݐ·1subscriptݑ’ݐ´ݑŸ1ݛ¼subscriptݑ£ݑ¦1subscriptݑ£ݑ¥1\fracv_x2v_x1=\frac(1-\alpha e_T)(1+\alpha)+\frac\alpha(1+e_T% )(1+\alpha)\left(\fracr\omega_1v_x1\right)-\fracD(1+e_A)r(1+% \alpha)\left(\fracv_y1v_x1\right)divide start_ARG italic_v start_POSTSUBSCRIPT italic_x 2 end_POSTSUBSCRIPT end_ARG start_ARG italic_v start_POSTSUBSCRIPT italic_x 1 end_POSTSUBSCRIPT end_ARG = divide start_ARG ( 1 - italic_α italic_e start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT ) end_ARG start_ARG ( 1 + italic_α ) end_ARG + divide start_ARG italic_α ( 1 + italic_e start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT ) end_ARG start_ARG ( 1 + italic_α ) end_ARG ( divide start_ARG italic_r italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG italic_v start_POSTSUBSCRIPT italic_x 1 end_POSTSUBSCRIPT end_ARG ) - divide start_ARG italic_D ( 1 + italic_e start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ) end_ARG start_ARG italic_r ( 1 + italic_α ) end_ARG ( divide start_ARG italic_v start_POSTSUBSCRIPT italic_y 1 end_POSTSUBSCRIPT end_ARG start_ARG italic_v start_POSTSUBSCRIPT italic_x 1 end_POSTSUBSCRIPT end_ARG ) (8)
and

ω2ω1=(α-eT)(1+α)+(1+eT)(1+α)(vx1rω1)-D(1+eA)r(1+α)(vy1rω1).subscriptݜ”2subscriptݜ”1ݛ¼subscriptݑ’ݑ‡1ݛ¼1subscriptݑ’ݑ‡1ݛ¼subscriptݑ£ݑ¥1ݑŸsubscriptݜ”1ݐ·1subscriptݑ’ݐ´ݑŸ1ݛ¼subscriptݑ£ݑ¦1ݑŸsubscriptݜ”1\frac\omega_2\omega_1=\frac(\alpha-e_T)(1+\alpha)+\frac(1+e_T% )(1+\alpha)\left(\fracv_x1r\omega_1\right)-\fracD(1+e_A)r(1+% \alpha)\left(\fracv_y1r\omega_1\right)\,.divide start_ARG italic_ω start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG start_ARG italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG = divide start_ARG ( italic_α - italic_e start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT ) end_ARG start_ARG ( 1 + italic_α ) end_ARG + divide start_ARG ( 1 + italic_e start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT ) end_ARG start_ARG ( 1 + italic_α ) end_ARG ( divide start_ARG italic_v start_POSTSUBSCRIPT italic_x 1 end_POSTSUBSCRIPT end_ARG start_ARG italic_r italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG ) - divide start_ARG italic_D ( 1 + italic_e start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ) end_ARG start_ARG italic_r ( 1 + italic_α ) end_ARG ( divide start_ARG italic_v start_POSTSUBSCRIPT italic_y 1 end_POSTSUBSCRIPT end_ARG start_ARG italic_r italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG ) . (9)
Eqs. 8 and 9, together with the definition of eAsubscriptݑ’ݐ´e_Aitalic_e start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT give a complete page_content of the scattering process in the sense that, for given initial conditions, there are three observables, vy2subscriptݑ£ݑ¦2v_y2italic_v start_POSTSUBSCRIPT italic_y 2 end_POSTSUBSCRIPT, vx2subscriptݑ£ݑ¥2v_x2italic_v start_POSTSUBSCRIPT italic_x 2 end_POSTSUBSCRIPT, and ω2subscriptݜ”2\omega_2italic_ω start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT, and three unknown parameters, eAsubscriptݑ’ݐ´e_Aitalic_e start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT, eTsubscriptݑ’ݑ‡e_Titalic_e start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT, and Dݐ·Ditalic_D, that can be inferred from a measurement of the observables. We have written Eq. 8 and 9 for the general case of nonzero Dݐ·Ditalic_D. However, as we will show in the next section, the present data are consistent with D=0ݐ·0D=0italic_D = 0, implying conservation of the ball’s angular momentum about the point of contact. The normal bounce speed of the ball is determined by eAsubscriptݑ’ݐ´e_Aitalic_e start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT, while for D≈0ݐ·0D\approx 0italic_D ≈ 0 the spin and tangential bounce speed are determined by eTsubscriptݑ’ݑ‡e_Titalic_e start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT and rω1/vx1ݑŸsubscriptݜ”1subscriptݑ£ݑ¥1r\omega_1/v_x1italic_r italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT / italic_v start_POSTSUBSCRIPT italic_x 1 end_POSTSUBSCRIPT. Depending on the magnitude and sign of the latter parameter, vx2subscriptݑ£ݑ¥2v_x2italic_v start_POSTSUBSCRIPT italic_x 2 end_POSTSUBSCRIPT and ω2subscriptݜ”2\omega_2italic_ω start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT can each be positive, zero or negative. Eqs. 8 and 9 are generalizations of equations written down by Cross[4] for the special case of the ball impacting a massive surface and reduce to those equations when eA=eysubscriptݑ’ݐ´subscriptݑ’ݑ¦e_A=e_yitalic_e start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT = italic_e start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT and eT=exsubscriptݑ’ݑ‡subscriptݑ’ݑ¥e_T=e_xitalic_e start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT = italic_e start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT.

IV Experimental results and discussion

A Determination of eTsubscriptݑ’ݑ‡e_Titalic_e start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT

We initially analyze the data using Eqs. 8 and 9 assuming D=0ݐ·0D=0italic_D = 0 and postpone for the time being a discussion of angular momentum conservation. Results obtained when the ball was incident on the bat without initial spin are shown in Fig. 2. The ball impacted the bat at speeds varying from 3.8 to 4.2 m/s but the results in Fig. 2 were scaled to an incident speed of 4.0 m/s by assuming that the rebound speed and spin are both directly proportional to the incident ball speed, as expected theoretically. An experimental value eA=0.375±0.005subscriptݑ’ݐ´plus-or-minus0.3750.005e_A=0.375\pm 0.005italic_e start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT = 0.375 ± 0.005 was determined from results at low (back) scattering angles, and this value was used together with Eqs. 8 and 9 to calculate rebound speed, spin, and scattering angle as functions of the impact parameter for various assumed values of eTsubscriptݑ’ݑ‡e_Titalic_e start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT. Best fits to the experimental data were found when eT=0subscriptݑ’ݑ‡0e_T=0italic_e start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT = 0 but reasonable fits could also be obtained with eT=0±0.1subscriptݑ’ݑ‡plus-or-minus00.1e_T=0\pm 0.1italic_e start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT = 0 ± 0.1.

Results obtained when the ball was incident with topspin or backspin are shown in Fig. 3. These results are not expected to scale with either the incident speed or incident spin and have not been normalized. Consequently the data show slightly more scatter than those presented in Figs. 2. The ball impacted the bat at speeds varying from 3.9 to 4.1 m/s. With topspin varying from 75 to 83 rad/s or with backspin varying from -72 to -78 rad/s. The ball impacted the bat at speeds varying from 3.9 to 4.1 m/s. With topspin varying from 75 to 83 rad/s or with backspin varying from -72 to -78 rad/s.9 to 4.1 m/s and with topspin varying from 75 to 83 rad/s or with backspin varying from -72 to -78 rad/s. Simultaneous fits to all three data sets resulted in eA=0.37±0.02subscriptݑ’ݐ´plus-or-minus0.370.02e_A=0.37\pm 0.02italic_e start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT = 0.37 ± 0.02 and with eT=0±0.02subscriptݑ’ݑ‡plus-or-minus00.02e_T=0\pm 0.02italic_e start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT = 0 ± 0.02. Using the recoil factors ry=0.188subscriptݑŸݑ¦0.188r_y=0.188italic_r start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT = 0.188 and rx=0.159subscriptݑŸݑ¥0.159r_x=0.159italic_r start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT = 0.159, our values for eAsubscriptݑ’ݐ´e_Aitalic_e start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT and eTsubscriptݑ’ݑ‡e_Titalic_e start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT imply ey=0.63±0.01subscriptݑ’ݑ¦plus-or-minus0.630.01e_y=0.63\pm 0.01italic_e start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT = 0.63 ± 0.01 and ex=0.16±0.02subscriptݑ’ݑ¥plus-or-minus0.160.02e_x=0.16\pm 0.02italic_e start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT = 0.16 ± 0.02. The result for exsubscriptݑ’ݑ¥e_xitalic_e start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT is consistent with that measured by impacting the ball onto a hard floor (0.17±0.03plus-or-minus0.170.030.17\pm 0.030.17 ± 0.03) but the result for eysubscriptݑ’ݑ¦e_yitalic_e start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT is slightly higher, presumably because of the lower impact speed and the softer impact on the bat. On the other hand, Figs. 2 and 3 show that the measured ω2subscriptݜ”2\omega_2italic_ω start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT values are inconsistent with eT=-0.14subscriptݑ’ݑ‡0.14e_T=-0.14italic_e start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT = - 0.14, which is the result expected for our bat if ex=0subscriptݑ’ݑ¥0e_x=0italic_e start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT = 0, as assumed by SHS.[1]

We next investigate the more general case in which angular momentum is not conserved by fitting the data to Eqs. 8 and 9 allowing both Dݐ·Ditalic_D and eTsubscriptݑ’ݑ‡e_Titalic_e start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT as adjustable parameters. Fitting to all three data sets simultaneously, we find eT=0±0.02subscriptݑ’ݑ‡plus-or-minus00.02e_T=0\pm 0.02italic_e start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT = 0 ± 0.02 and D=0.21±0.29ݐ·plus-or-minus0.210.29D=0.21\pm 0.29italic_D = 0.21 ± 0.29 mm. This justifies our earlier neglect of Dݐ·Ditalic_D and confirms that the data are consistent with angular momentum conservation. All calculations discussed below will assume D=0ݐ·0D=0italic_D = 0.

It is possible to determine the incident and outgoing angles with respect to the normal, θ1su

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