BERRY, J.

A jury convicted the defendant of rape and assault and battery. On appeal, the defendant assigns as error (1) the denial of his motion to suppress items seized in a search of the defendant's Jeep automobile and truck; (2) the admission of certain testimony that, it is contended, violated the first complaint doctrine; and (3) aspects of the judge's instructions, including instructions on first complaint and the elements of the offense of rape. We affirm.

1. Background. We summarize the trial evidence, reserving further details that relate exclusively to the suppression motion. In the summer of 2004, the defendant and the victim met while working out in a gym located in Gardner. At some point, the two engaged in conversations concerning the victim's interest in finding a job as a correction officer. The defendant offered the name of a State senator who, he said, might be of some help in her search. (That the defendant's offer had some prospect was supported by the defendant's familiarity with law enforcement employment in that the defendant held a position with the county house of correction.) The defendant provided the victim with the senator's card. She called the senator's office and an interview was ultimately arranged for November 8, 2004.

On the evening of November 3, 2004, the victim arrived at the gym at approximately 7:30 P.M. The defendant was finishing his workout regimen. He complimented the victim's appearance and asked if she had met with the senator; she replied that her meeting was scheduled for the following Monday. This brief exchange ended and they went their separate ways.

Approximately one hour later, the victim left the gym and was walking toward her car. The defendant was outside, standing next to his red Jeep, which was parked beside her car. He asked her to go out for a coffee. When she declined, he explained that it would be a quick chat about her upcoming meeting. She then agreed to ride with the defendant to go get coffee. When they were some distance from the gym, the victim inquired where they were going. The defendant replied that he was going to "show you where you will be meeting" the senator. He drove for twenty minutes, before reaching the Barre town hall, which he identified as the site for her interview. The defendant then left the town center and proceeded on a dirt road, bordered by a heavily wooded landscape, assuring the victim that this route was a shortcut to a main roadway. (As shall be seen in reference to the first complaint issue, the location of the secluded areas where the defendant took the victim, how one would reach these places, and whether the victim could have just walked away from the defendant were contested trial issues.)

Passing by a cemetery in an otherwise desolate area, the defendant stopped the Jeep. A struggle ensued. The defendant took out a handgun and handcuffs from the glove compartment. (The gun also was a subject of what the defendant contends was improper first complaint testimony, and the handcuffs were the subject of the suppression motion; both issues are discussed herein.) Fighting him off, the victim tried to escape from the Jeep, but the defendant grabbed her hair and jacket. The defendant apologized, stated he would take her back to her car, and expressed regret that he "didn't mean for it to happen that way." As he drove, he kept the gun close to his body or in his right hand.

Thirty minutes later, while still on back roads in another densely wooded and sparsely populated area, the defendant again brought the Jeep to a halt. At another secluded spot with a fire pit close by, the defendant ordered the victim to move into the back seat. The defendant joined her, pulled down his trousers, and inserted his penis in her vagina. The defendant then tried but was unable to penetrate her anally. He kept rubbing his penis on her until he ejaculated. Once he did, he pulled up his trousers and returned to the driver's seat. The victim put her clothes on and got into the front passenger seat. At some point, the victim gained possession of the gun. They reached the gym parking lot. The defendant threatened to kill the victim if she told anyone what had happened. She fled the Jeep. Before driving away, she placed the handgun back inside the Jeep through its open passenger door. (At some point during the events described above, the defendant did not have the gun in hand or close to him and the victim had gained possession of it.) She drove to the Gardner police station. She ran toward an officer outside — Sergeant Roger Wrigley — screaming that she had been raped. She described certain details, including the defendant's name. The victim was taken to a hospital for treatment.

The theory of the defense was consensual sex. The defendant testified. He indicated that the victim had made sexual overtures to him in the past and on the evening in question. The defendant stated that the sex was with the victim's consent.

2. Suppression motions. The defendant filed motions to suppress directed at certain statements the defendant made during a police interview, a gun and clothing seized from his house, and items that were within the red Jeep and a truck parked in the defendant's driveway on the night of his arrest outside the house. The particular items at issue, which were seized from the vehicles, are a clump of hair and a pair of handcuffs, which were observed within the truck by the police and later were seized pursuant to a warrant, after the vehicles were taken to the police station.

The suppression motions were the subject of an evidentiary hearing. The motion judge suppressed the defendant's statements during the police interview, finding that there was a failure by the police to abide by the defendant's request to end questioning. The judge also suppressed a gun and clothing found in the defendant's house, on the basis that the police entry into the house, which was with the wife's consent, was tainted by the improper questioning and information obtained during the defendant's police interview. The judge denied the motion to suppress with respect to the hair and handcuffs seized in the vehicles. We turn to this last suppression ruling, which the defendant now challenges on appeal.

The subsidiary findings of fact by the motion judge, which the defendant does not challenge, in summary form, are as follows. Late in the evening of November 3, 2004, commencing at about 11:00 P.M., police dispatches issued concerning a sexual assault and the use of a firearm, which had occurred earlier that evening. The dispatches referenced a red Jeep "SUV" automobile, stated a license plate number, and identified the suspect as the defendant Starkweather, residing at a particular address in Barre.

Officer Ury and Officer Holgerson separately drove to the Starkweather home address and arrived at approximately 11:20 P.M. Officer Ury observed a man in the driveway, standing between two parked vehicles: a red Jeep and a pickup truck. Officer Ury testified that the man was going from the Jeep to the truck, and the driver's side doors of both were open. It appeared that he was moving things between the two vehicles. The man in the driveway identified himself as John Starkweather. The officers approached, with guns drawn, and arrested the defendant.

Officer Ury handcuffed the defendant, gave Miranda warnings, and placed the defendant inside the police cruiser. Moments later, with the aid of flashlights, the two officers reached into the passenger compartments of the Jeep and the truck, to see if there was gun therein, as the victim had described. No gun was seen, but the officers did see a pair of handcuffs on the truck's gear box. Nothing was removed, and both vehicles were impounded and towed to the Gardner police garage, where a search was conducted pursuant to a warrant.

In his appeal, the defendant argues that the motion judge erred, as matter of law, in ruling that the vehicle searches were constitutionally permissible, as incident to the defendant's arrest, and that, notwithstanding a warrant's subsequently being obtained after the initial flashlight view and reach into the vehicles' interiors, a taint remained which was not cured by the later issued warrant. We disagree on two legal fronts. We conclude that the warrant exception for a search incident to arrest applies, and that the evidence would inevitably have been discovered.

First, we agree with the motion judge that this was a valid search of the defendant's Jeep and truck incident to his arrest because there was probable cause that evidence relative to the offenses at issue would be contained in the two vehicles. Such a warrantless probable cause search of a vehicle is a longstanding and firmly rooted exception to the warrant requirement. See Weeks v. United States, 232 U.S. 383, 392 (1914) (search of person when legally arrested to discover and seize the fruits or evidence of the crime long recognized).

The defendant, citing Arizona v. Gant, 129 S. Ct. 1710 (2009) ( Gant), contends, however, that after he was arrested, handcuffed, and placed in the cruiser, he could not gain access to or exercise control over the passenger compartments of either vehicle, and, that, accordingly, under the Gant case, the search of his vehicles as incident to his arrest was not constitutionally justified under the United States Constitution. That is not what Gant held.

In Gant, the United States Supreme Court held that, "after the arrestee has been secured and cannot access the interior of the vehicle," 129 S. Ct. at 1714, the search-within-reach distance rule of Chimel v. California, 395 U.S. 752, 763 (1969) ( Chimel), no longer applies, and there is no other constitutional justification for a warrantless search of the entire vehicle interior or containers therein. However, of import, and contrary to the defendant's contention, the Gant case did not disturb the constitutional search and seizure law, which allows for a warrantless vehicle search where there is probable cause that the subject vehicle may contain evidence of the crime for which the arrest is made. Gant, 129 S. Ct. at 1721. Moreover, as Gant recognizes, such a probable cause vehicle search right continues even after the arrestee is taken away from the vehicle and is secured. On these points of search and seizure law, the Supreme Court in Gant held that the "circumstances unique to the vehicle context justify a search incident to a lawful arrest when it is 'reasonable to believe evidence relevant to the crime of arrest might be found in the vehicle.'" Gant, 129 S. Ct. at 1719, quoting from Thornton v. United States, 541 U.S. 615, 632 (2004) ( Thornton) (Scalia, J., concurring in judgment). "If there is probable cause to believe a vehicle contains evidence of criminal activity, United States v. Ross, 456 U.S. 798, 820-821 . . . (1982), authorizes a search of any area of the vehicle in which the evidence might be found." Gant, 129 S. Ct. at 1721. See Commonwealth v. Young, 78 Mass. App. Ct. 548, 550 n. 3, 554-555 nn. 7 8 (2011).

In this case, the defendant's arrest was precisely for the crime that was the predicate for the search. See G.L. c. 276, § l.

The defendant was arrested for having committed a felony sexual assault offense, which the arresting officers, acting with probable cause, could reasonably believe occurred that same evening. Based on the radio dispatches, the officers could reasonably believe that the suspect might be armed, and that the Jeep where the rape took place and the truck — to which the defendant appeared to be moving things — probably would harbor evidence relevant to the crime.

Second, even if the search was not justified as incident to arrest, the doctrine of inevitable discovery would apply. The information in the search warrant affidavit was sufficient to justify issuance of the warrant, even apart from the original flashlight sighting of the handcuffs and the hair in the truck. (The defendant, prior to his arrest, appeared to be moving things between the red Jeep, where the assault had taken place, and the truck, both of which were parked in the driveway at the time of the arrest.) "[I]f the government can prove that the evidence would have been obtained inevitably and, therefore, would have been admitted regardless of any overreaching by the police, there is no rational basis to keep that evidence from the jury in order to ensure the fairness of the trial proceedings. In that situation, the State has gained no advantage at trial and the defendant has suffered no prejudice." Nix v. Williams, 467 U.S. 431, 447 (1984). "Under art. 14 of the Massachusetts Declaration of Rights, the Commonwealth satisfies the inevitable discovery exception to the exclusionary rule if it proves by a preponderance of the evidence that 'discovery by lawful means was certain as a practical matter.' " Commonwealth v. Linton, 456 Mass. 534, 558 (2010), quoting from Commonwealth v. O'Connor, 406 Mass. 112, 117 (1989).

3. First complaint doctrine. The defendant argues that there were multiple instances in which the testimony — particularly of police officers — strayed beyond the permissible boundaries of first complaint evidence set forth in Commonwealth v. King, 445 Mass. 217, 241-247 (2005), cert. denied, 546 U.S. 1216 (2006). There was no objection. Accordingly, we review to determine whether there was error under the first complaint doctrine and, if there was such error, whether the introduction of the evidence yielded a substantial risk of a miscarriage of justice. Commonwealth v. Freeman, 352 Mass. 556, 563-564 (1967). Commonwealth v. McCoy, 456 Mass. 838, 850 (2010). We conclude that most of the challenged statements did not violate the first complaint doctrine, and that the few testimonial statements which did go beyond the doctrine did not give rise to a substantial risk of a miscarriage of justice.

We apply the analysis in Commonwealth v. Arana, 453 Mass. 214, 225-226 (2009), and Commonwealth v. Dargon, 457 Mass. 387, 400 (2010), to determine whether the challenged evidence was independently admissible, served some purpose other than to corroborate the victim's accusation, and was sufficiently important to a fair understanding of the Commonwealth's case so that the probative value of the evidence outweighed any prejudice. Having done so, we conclude that the challenged testimony was, in large part, independently admissible proof of the victim's demeanor, her state of mind, and the manner and means of the police investigation.

Generally, the evidence which the defendant contests involves the following: a few references in the testimony of the designated first complaint witness, Sergeant Wrigley, that a recorded statement was taken in the emergency room and later a written statement was taken at the victim's house (in these brief references, there was no detail about the substance of the statements); a comment by the victim to the police that the gun held by the defendant looked like the gun an officer carried; a reference by a police officer to what the victim described as a clump of hair pulled from her head being subject to analysis; a comment by the victim that she did not leave the Jeep during the attack, which statement was offered to rebut a defense point about there being no footprint evidence; and a few comments by the victim confirming locations, as she drove with the police officers to identify and locate the secluded area where the rape transpired. We summarize briefly the testimony challenged under the first complaint doctrine.

a. Sergeant Wrigley's testimony. The defendant claims that Sergeant Wrigley, the designated first complaint witness, improperly described a comment that the victim had made while Sergeants Wrigley and Dale were with her at the hospital. This comment was to the effect that the gun which Sergeant Dale carried, a Glock brand handgun, appeared similar to the gun the defendant had possessed during the attack. We believe this comment had independent significance concerning the identification of the gun that was present in the defendant's vehicle. Thus, we see no first complaint violation. See Commonwealth v. Arana, 453 Mass. at 224-227, 229; Commonwealth v. Dargon, 457 Mass. at 399-400. b. Sergeant Watson's testimony. The defendant contends that Sergeant Watson, who was present at the hospital, improperly referred to a recorded statement that Watson took from the victim while she was in the emergency room. The recounting of a victim's additional statements beyond the initial statement to the designated first complaint witness is improper evidence. Commonwealth v. Stuckich, 450 Mass. 449, 457 n. 11 (2008). Commonwealth v. McCoy, 456 Mass. at 846. But, in this trial, the officer's references to the victim giving such a recorded statement were offered in connection with a description of the victim's demeanor. "Evidence of a victim's state of mind or behavior following a crime has long been admissible if relevant to a contested issue in a case." Commonwealth v. Arana, 453 Mass. at 225. Demeanor evidence may be of particular importance in a case such as this, where the trial devolves to a contest of credibility concerning whether the charged offense, here rape, occurred. See id. at 225-226. In our view, Sergeant Watson's firsthand observations of the victim's demeanor in the emergency room, and physical and emotional state of mind in the wake of the sexual attack, played an integral role in assessing credibility. The reference to a recorded statement in this demeanor-related context was not error under the first complaint doctrine.

The defendant also takes issue with Sergeant Watson's statement that he traveled to the victim's house following the emergency room treatment and took an additional statement to clarify certain points. We discern no independent basis for this testimony other than to cite a successive complaint, which is contrary to the first complaint doctrine. However, the reference was so fleeting that we see no substantial risk of a miscarriage of justice presented.

c. Crime location statements. The defendant argues that the testimony of Sergeant Watson and Officer Duplessis concerning the drive in which the victim accompanied the police to locate the crime scene — which took hours of backtracking and retracing circuitous paths and secluded roads — included references that violated the first complaint doctrine. We disagree. This evidence, in general, was admissible to rebut the defendant's claim that the victim could easily have fled from the Jeep and walked away. Furthermore, the few isolated references to remarks by the victim were in the context of demeanor evidence and occurred as the prosecutor asked how the victim reacted when she recognized certain places. Indeed, virtually all the contested references included a link to such demeanor and state of mind evidence and included descriptions such as "she then began to remember exactly where she was, and as we went by the dirt road on Route 62, her demeanor changed completely. . . . She just became animated, from being very soft and quiet spoken." For the reasons previously stated, this evidence had independent significance. See generally Commonwealth v. Arana, 453 Mass. at 225-226. See also discussion in Part 3(b), supra.

Beyond that, this evidence of the police car ride to find the crime scene was independently admissible to describe the course of the police investigation. As the Supreme Judicial Court stated, "We did not intend in Stuckich to imply that, unless a police officer is the first complaint witness, testimony concerning the circumstances giving rise to the police involvement in a sexual assault case will never be admissible as part of the Commonwealth's case-in-chief." Commonwealth v. Arana, 453 Mass. at 226. Indeed, the Arana court confirmed that both the "circumstances" and "timing" of police involvement may be admissible and relevant as a response to the countervailing defense theory of the case. See id. at 227.

d. Testimony of Sergeant Lyver. The defendant claims that during redirect examination of Sergeant Lyver there were two first complaint violations. The first reference arose in the context of the sergeant stating that he had supervised the chemist who initiated tests on a hair clump found in the defendant's vehicle, which the victim had stated was pulled from her head during the attack. The second reference involved the sergeant's observations at the crime scene concerning the absence of footprints; this was connected to a reference to the victim's statement that neither she nor the defendant had exited the Jeep while it was parked at the crime scene. Both of these references had, we conclude, independent significance (1) in connecting the clump of hair pulled from the victim's head with that found in the defendant's truck, and (2) in providing an explanation that neither the victim nor the defendant had exited the truck, so that the footprint evidence would not exist. Thus, the police investigation analysis articulated in Arana, 453 Mass. at 226-227, is implicated, as discussed above, and these two references were not in violation of the first complaint rule.

e. The victim's testimony. The defendant says the victim improperly referred to other statements she had given to Sergeant Watson. (See discussion in Part 3[b], supra.) As Stuckich pronounced, a victim "should not have been allowed to testify on direct examination" about statements other than those which qualify as statements to the first complaint witness. Commonwealth v. Stuckich, 450 Mass. at 457. Given the level of detail set forth in all of the victim's other, properly admitted testimony, and that of the other witnesses, we do not see that these passing references gave rise to a substantial risk of a miscarriage of justice. (With respect to the victim's identification of the places of the offense during the police ride, we have already addressed such comments.)

In sum, we conclude that most of the testimony the defendant challenges had independent significance and did not violate the first complaint doctrine. With respect to the few references that strayed beyond the confines of the first complaint doctrine, we conclude that there was not presented a substantial risk of a miscarriage of justice. We note that the jury acquitted the defendant of the other serious charges of aggravated rape, assault with intent to murder while possessing a firearm, kidnapping, assault and battery by means of a dangerous weapon, and indecent assault and battery. The rape conviction was returned as a lesser included offense of the aggravated rape indictment. These acquittals and the conviction on the lesser included offense support our determination that there was no substantial risk of a miscarriage of justice.

4. The jury instructions. The defendant claims several errors in the judge's jury instructions. No objections were registered. We conclude there was no error in the challenged instructions.

The defendant maintains that the trial judge erred by administering an ambiguous limiting instruction that allowed the jury to consider, for substantive purposes, an unspecified portion of the victim's dialogue with Sergeant Wrigley, the designated first complaint witness. The record does not support this contention. The trial judge correctly observed the timing and procedure for designating a first complaint witness by administering a limiting instruction "contemporaneously with the first complaint testimony, and again during the final instructions." Commonwealth v. King, 445 Mass. at 248. Specifically, the trial judge instructed the jury to consider the victim's "initial statement" to Sergeant Wrigley "for any purpose" and that the "remainder of [her statement to Sergeant Wrigley]" (the first complaint testimony) was to be considered for a "limited purpose."

Further, and contrary to the defendant's contention, the Commonwealth is not required under G.L. c. 265, § 22(6), to prove in a rape case that the defendant lacked a reasonable and good faith belief as to the victim's consent. Commonwealth v. Urban, 450 Mass. 608, 615 n. 5 (2008). See Commonwealth v. Caracciola, 409 Mass. 648, 652-653 (1991).

During deliberation, the jury asked as follows. "Does the defendant have to know the sexual intercourse was against the complainant's will?" The judge appropriately answered this question in a supplemental instruction. The defendant did not object at trial. On appeal, he argues that the trial judge erred in failing to answer the jury's question with a simple "yes." That contention is without merit. "The necessity, extent, and character of supplemental instructions in response to a jury request are matters within a trial judge's discretion." Commonwealth v. Johnson, 429 Mass. 745, 753 (1999), quoting from Commonwealth v. O'Connor, 407 Mass. 663, 667 (1990). The defendant also contends that the supplemental instruction failed to adequately define the term "constructive force." This contention is not correct. In this supplemental instruction, as in the original jury charge, the terms physical force and constructive force were appropriately defined. "[I]n determining whether a jury instruction is defective, we view the charge in its entirety and consider how it likely would have been interpreted by a reasonable juror in the over-all context of the instructions." Commonwealth v. Manzelli, 68 Mass. App. Ct. 691, 697 (2007). Judgments affirmed.